Y., under the act of August 24, 1912. Acceptance for mailing at 
special rate of postage provided for in section 1103, act of yee a 
( cman’ 3, A917; authorized July 19, 19138 


~ Published Fortnightly 


ntered as bine chai, ieee Atigaok 2, 1913, ‘at the Post Office. at Albany, i) 


"ALBANY, Nai 


THE UNIVERSITY OF THE STATE OF NEW YORK 


Regents of the University 
With years when terms expire 


1934 Cuxster S. Lorp M.A., LL.D., Chancellor - - Brooklyn , 
1936 ApELBERT Moot LL.D., Vice Chancellor - - ~- Buffalo 
1927 ALBERT VANDER VEER M.D., M.A., Ph.D., LL.D. Albany 
1937 Cuartes B. ALEXANDER M.A., ey LUD. 


Eth, ei eee ee - - - - - Tuxedo 
1928 Water Guest Kettoce B. LL. D. - - - + Ogdensburg 
1932 James Byrne B.A., LL.B., LL.D. - - - - - New York 
1931 Tuomas J. Mancan M.A. - - - - - - - Binghamton 
1933 Witttam J. Wattin M.A. - - - - - - Yonkers 


1935 Witt1am Bonpy M.A., LL.B., Ph.D., D. C. L. - New York 
1930 Wititam P, BAKER BL. ne D. - - - - - Syracuse 
1929 Ropert W. Hicpie M.A. - - - - - - - - Jamaica 
1926 Rotanp B. Woopwarp B.A.- - - - - = = Rochester 


sd 


President of the University and Commissioner of Education 


FRANK P. Graves Ph.D., Litt. D., L.H.D., LL.D. 


Deputy Commissioner and Counsel 


FRANK B, Gitpert B.A., LL.D. 


Assistant Commissioner and Director of Professional Education 


Avueustus S. Downtne M.A., Pd.D., L.H.D., LL.D. 


Assistant Commissioner for Secondary Education 


James Suttivan M.A., Ph.D. 


Assistant Commissioner for Elementary Education 


Greorce M. Witey M.A., Pd.D., LL.D. 


Director of State Library 


James I. Wver M.L.S., Pd.D. 


Director of Science and State Museum 


Joun M. Crarxe Ph.D., D.Sce., LL.D. 


Directors of Divisions 


Administration, Ltoyp L. CHEenry B.A. 

Archives and History, ALEXANDER C, LICE M. AY Late Ds PRD: 
Attendance, JAMES D. SULLIVAN 

Examinations and Inspections, Avery W. SKINNER B.A., Pd.D. 
Finance, CLARK W. HALLIDAY 

Law, Inw1In Esmonp Ph.B., LL.B. 

Library Extension, Witt1am R. Watson B.S. 

School Buildings and Grounds, FRanK H. Woop M.A. 

Visual Instruction, ALFRED W. ABRAMS Ph.B. 

Vocational and Extension Education, Lewis A. WiILson 


FOREWORD 


This outline for the study of arithmetic in the elementary grades 
ei 


; i bbs been prepared under the direction of the Commissioner of Edu- 
_ cation by the following committee: Dr H. DeW. DeGroat, principal, 
- State Normal School, Cortland; chairman; Frances Killen, super- 
visor of Elementary grades, Dunkirk; and Ina D. Porter, classroom 
ia Be Schenectady. This committee has been greatly assisted by 
the suggestions of successful teachers and supervisors throughout 
this and other states. At the request of many teachers some method 
work has been included. The suggestions for the work covering 
: _ the seventh and eighth years recognize that the point of view neces- 


sary for successful work in these higher grades is different from 


ae very helpful to he teachers of arithmetic throughout the State. 
GrorceE M. WILEY 
Assistant Commissioner for Elementary Education 


‘al that of the lower grades. It is the hope that this outline will prove — 


TABLE OFF CONTENTS 


PAGE 
General introduction 233.505) in. Pelee oe bee oe ecaeel: ee 5 
First grade: 30. 34 22205 sia ia teenie « a aol © aoe oe aie a aa ee 13 
Second grade ss lic) 56 eis odes ates ore sln sO eaed ata ele Tin cee Och 24 
Third grade oi. sess Fs fu sth ae ete See ee 42 
Fourth: grade so. 3 5 os sic ee eine sce eels a Sas ciate ole lel eta ee ee cae 56 
Putth ¢ Prades yes singh lek oleate cn Seo bse in veretor aie aheeta ie cakes Pat a tate en 72 
Sixth  gradep nis vases cern €0cs aie eee toe Mn BA clea © ne es eee ee ee 82 
Introduction ‘to arithmetic in upper grades. 22. %o..s sie sca eee eee 91 
Seventh: terddes.t. oyind Arai ee Sa eee Whe le hee le eel oe Ld 93 


Eighth -grad@ cif sitio 2 < eiatae siete sce ates salen any te 104 


University of the State of New York Bulletin 


Entered as second-class matter August 2, 1913, at the Post Office at Albany, 
N. Y., under the act of August 24, 1912. Acceptance for mailing at 
special rate of postage provided for in section 1103, act of 
October 3, 1917, authorized July 19, 1918 


Published Fortnightly 


No. 815 ALBANY, N.. ¥: November 1, 1924 


Syllabus for Elementary Schools 


ARITHMETIC 


GENERAL INTRODUCTION 


In the opening chapter to Thorndike’s The Psychology of Arith- 
metic! is the following statement regarding the functions of the 
elementary grades in the teaching of arithmetic: 

“What are the functions that the elementary school tries to 
improve in its teaching of arithmetic?” Other matters might well 
be considered in this connection, but the main outline of the work 
of the elementary school is now fairly clear. The arithmetical func- 
tions or abilities which it seeks to improve are, we may say: 

(1) Working knowledge of the meanings of numbers as names 
for certain sized collections, for certain relative magnitudes, the 
magnitude of unity being known, and for certain centers of nuclei 
of relations to other numbers. 

(2) Working knowledge of the system of decimal notation. 

(3) Working knowledge of the meanings of addition, subtraction, 
multiplication, and division. 

(4) Working knowledge of the nature and relations of certain 
common measures. 

(5) Working ability to add, subtract, multiply, and divide with 
integers, common and decimal fractions, and denominate numbers, 
all being real positive numbers. 

(6) Working knowledge of words, symbols, diagrams, and the 
like as required by life’s simpler arithmetical demands or by econom- 
ical preparation therefor. 

(7) The ability to apply all the above as required by life’s simpler 
arithmetical demands or by economical preparation therefor, includ- 
ing (7a) certain specific abilities to solve problems concerning areas 


1 Thorndike. The Psychology of Arithmetic. Macmillan. p. 23. 


6 THE UNIVERSITY OF THE STATE OF NEW YORK 


of rectangles, volumes of rectangular solids, per cents, interest, and 
certain other common occurrences in household, factory, and busi- 
ness life. 

It may be reasonable to assume that teaching of arithmetic in the 
elementary grades has at times failed of its purpose; first, because 
of the well-meaning but unfortunate attempt on the part of school 
authorities to include topics far beyond “ life’s simpler arithmetical 
demands,” second, through the failure on the part of the school 
program to give the necessary “attention to perfecting the more 
elementary abilities.” 

In the organization of the work presented herewith as an outline 
for the elementary grades the committee has been guided in large 
part by these principles as determining the general aims to be 
attained. Arithmetic that is taught purely as a mechanical procedure 
unrelated to social needs fails to function. This has been clearly 
demonstrated in much of our mathematics teaching. 

Arithmetic must be taught, not as an abstract subject independent 
of other interests, but as immediately and vitally related to all com- 
mon activities and interests of everyday life. 

This difference in emphasis between the work in the first 6 years 
of the elementary course and the work of the last 2 years is nowhere 
expressed better than by Klapper in his The Teaching of Arithmetic.* 
In the discussion of the aims of arithmetic for these two levels of 
the school course are found the following statements: 

Aim of the first six years: To develop skill in the fundamental 
operations as applied to whole numbers, common and decimal frac- 
tions, and to teach the necessary number facts. 

Aim of last two years: To develop power to understand arith- 
metical solutions and manipulate quantities in problems varying from 
the type. 

Not only are the aims of the work in these two periods clearly 
expressed but there is also much to be gained from the emphasis 
that is given to the importance of recognition of definite aims and 
objectives for the course. It is stated further in this connection 
regarding courses of study that “ There is little indication that those 
charged with the making of the course first formulate the aims of 
the course in arithmetic as an intelligent basis for selection of 
material.” The importance of such a step in the organization of 
any course of study can not be overemphasized. 

There has been no attempt to introduce the elements of algebra 
or geometry as such into the course in the higher grades. The 
algebraic elements that appear and the simple problems of mensura- 


1 Klapper. The Teaching of Arithmetic. Appleton. p. 24. 


ELEMENTARY ARITHMETIC SYLLABUS fi 


tion involving geometric forms have been introduced as integral 
parts of a well-balanced course in elementary mathematics. The use 
of literal elements and the interpretation of simple formulas are 
used in the syllabus outline as‘a definite part of the suggested course 
of study in arithmetic. 

If there is any thought on the part of local school authorities in 
any community that they wish to introduce elementary algebra at 
the beginning of the second half of the eighth year, the work in 
arithmetic should be brought to a close at that point and the new 
subject of algebra introduced independently. 

At the very beginning of the course the teacher should make use 
of the experience of the child gained from his environment previous 
to his entrance to school. The school life itself and the pupil’s 
new experiences in his rapidly widening horizon will provide much 
interesting material for this purpose. With some modifications this 


‘principle may well determine the procedure throughout the course 


in arithmetic. The teacher can and should so modify or add to the 
pupil’s experiences in number relationships and in the practical appli- 
cations of arithmetic to life as to secure a constant reaction of 
environment on the entire subject of arithmetic—the one thing in 
teaching tending most to vitalize the subject taught. 

It is not the thought that this syllabus should be used as an 
inflexible guide. It has been prepared rather as a suggestion in the 
hope that it will prove helpful and inspiring to teachers and will not 
be interpreted as a mere outline of topics. In some instances methods 
of presentation have been suggested. A variety of material has 
been presented that may be used or modified as may be determined 
by local school authorities. While some topics have been made 
entirely optional, others have been strongly recommended. There 
is therefore every opportunity for the use of the syllabus in the 
communities throughout the State with varying emphasis on different 
portions of the work. 

The committee has assumed that the primary purpose of arithmetic 
to be realized at the end of the sixth school year is to furnish the 
pupil with the tools with which to work, that is, a mastery of funda- 
mental processes in integers, fractions and decimals. In contrast 
with the work for the first 6 years, the emphasis in the plan for the 
seventh and eighth grades is on the application of these fundamental 
processes to the problems of home, business and community life. 

In the allotment of time to be given to the subject of arithmetic 
in each of the 8 years of the elementary course the practice will vary 
in different school systems. The committee has hesitated to make 


8 THE UNIVERSITY OF THE STATE OF NEW YORK 


any recommendation on this point because of varying conditions in 
different localities. In the highly organized school system ample 
allotments can be given for this subject as well as for others through- 
out the grade. On the other hand in the small rural school where 
one teacher carries the entire responsibility for the elementary school 
program, grades must be handled in groups, and a reasonable time 
allotment is therefore much more difficult to meet. 

The following average number of minutes each week given to 
the subject of arithmetic in the elementary grades is taken from 
recently submitted programs in the cities of the State: 


Average number Average number 


of minutes of minutes each 
given weekly week from Doctor 

Grade toarithmetic! Holmes’ study? 
Le aaa etek Re TT SRO er ey eee 94 93 
A PET On Ee ed PE MEY Pree LS /; 149 
1 vad. Sopher Sah Pee Ss eee ee ee eee 195 203 
ZS Te ORO DOE. od APN, TMNT A ATE ZS PS | 
CB Ss PAP eA etd Baile ine Pa es 088 216 Papa 
OU oeleane seek es aerae warte' st een ere 218 226 
LSE ONES lee ae Cn ee 220 ZA 
Sia tSasitaial gin Meese erst ken aaterte Rae «lett es 220 


Much of the work below the fourth grade will undoubtedly be oral. 
It is suggested that beginning with the fourth grade from 20 per 
cent to 25 per cent of the usual recitation period should be devoted 
to oral practice which might well include drill both on abstract work 
and on problem solution. 


The Solution of Problems 


In the lower grades all problems should deal with things with 
which the pupil is entirely familiar. Problems relating to things 
which the child possesses or which he may see every day should 
therefore be selected. A problem dealing with something familiar 
and near at hand is far easier than the same problem dealing with 
something that is strange or far away. 

The pupil should always understand clearly the language of the 
problem before he attempts to solve it. He should so thoroughly 
understand the situation that he can set it forth clearly in his own 
language. This he should frequently be required to do before 
attempting a solution. 

In the upper grades the data of the problem should be set forth 
by the pupil, and he should also be required to state what he is to 


1 Twenty-three New York cities. ; 
3 The Fourteenth Yearbook, National Society for the Study of Education, p. 21-27. 


ELEMENTARY ARITHMETIC SYLLABUS 9 


find. He should also name the unit in which his answer is to be 
given. If the pupil can set forth the conditions of the problem up 
to this point he will be well prepared to undertake the solution. 

Frequently the teacher should require the pupil to state clearly in 
advance, step by step, how he proposes to solve the problem. It is 
also good practice for pupils to state how they would solve certain 
problems without having numerical data given. For instance, what 
would be the cost of a certain number of oranges at a certain price 
per dozen? The pupil would be expected to give the two processes 
involved in the solution. 

Pupils should be given much practice in making original problems 
both orally and in writing. The written problems will offer an 
excellent exercise in English. In order to make problems, pupils 
should have an intimate acquaintance with the vocabulary involved. 
This means not simply a knowledge of definitions, but facility to 
use terms with a full understanding of them. 

After an answer has been obtained, the pupil should frequently 
be required to determine for himself whether the answer is reason- 
able, first by checking up with his own knowledge of units of cost, 
or distance, or weights etc., second, by approximating the answer. 

The process of approximation is one in which the pupil should 
have much drill, and the checking of answers is a subject to which 
the teacher should give much attention. 

The problem should not be left until the pupil has made some 
effort by approximation, by proof, by a different method of solution, 
or by a repetition of the solution to satisfy himself that his answer 
is correct. 

The teacher should inspire his pupils with a zeal for correct 
answers, which will not leave them satisfied with anything less than 
correct and accurate work. This zeal may be fostered by giving 
deserved praise for originality shown in thinking and in working 
problems. The greater the variety in the solutions by members of 
the class, the greater satisfaction should the teacher feel and express. 

Too often both teachers and pupils approach a series of miscel- 
laneous problems with dread. Problems should be and can be made 
a pleasure if the teacher in the assignment, by a little wise explana- 
tion, anticipates the greatest difficulties, requires some systematic 
reasoning process, praises effort and encourages originality. Above 
all, teachers should remember that there can be no success in the 
teaching of problems, unless pupils are required to make conscientious 
preparation. 


10 THE UNIVERSITY OF THE STATE OF NEW YORK 


The teacher should always feel free to omit from a list of prob- 
lems contained in a textbook any that are too difficult and any that 
may be poorly suited to the immediate purposes. 

Pupils often flounder in two-step problems in fractions simply 
because they are plunged into them with very little practice 
in the solution of two-step problems in integers. If a teacher finds 
that a class is weak in reasoning, he should make a close study to 
determine the reason. It may be because of a lack of practice in 
the solution of type problems. It may be because the pupil is so 
uncertain in his mechanical processes that he has little available 
mental power left to cope with the real difficulties of the situation. 
Often the explanation is the pupil’s inability to read. 

Whatever the cause of weakness in problem work, it is the 
teacher’s first responsibility to find it out. If the cause is lack of 
practice or mechanical inaccuracy, much easy practice will tend to 
correct the situation. Sometimes a set form of analysis may help. 
If the cause is the pupil’s inability to read, let the class in reading 
now and then use a textbook in arithmetic. Some skilful instruction 
in “silent reading ” ought to help the situation materially. 

Above all, the teacher should find poor work in the solution of 
problems a challenge to his best endeavor. A teacher enthusiastic 
about. his work in arithmetic is sure to find his enthusiasm met by 
both a like enthusiasm and a steady improvement on the part of 
his pupils. 

Tests 

It is recommended that to check up progress made and to deter- 
mine just what and how much instruction is needed in individual 
cases, standard tests be given from time to time. Printed instruc- 
tions for scoring and interpreting results should be ordered with 
the tests. 

Buckingham Scale for Problems in Arithmetic, Public School 

Publishing Co., Bloomington, II. 

The Courtis Research Tests, Series B (for tests covering the 

fundamental operations), Detroit, Mich., or similar tests 

Stone’s Standardized Reasoning Test in Arithmetic, Teachers 

College, Columbia University, New York City 

Woody-McCall’s Mixed Fundamentals, Teachers College, Colum- 

bia University, New York City 


Supplementary Material 


Flash cards or perception cards from any dealer in school supplies. 
More elaborate ones can be made. 


ELEMENTARY ARITHMETIC SYLLABUS 11 


Number boxes 

Play money 

Kindergarten sticks 

Peg boards 

Weights and measures, weighing scales (from local dealers) 

Fassett Number Cards, Milton Bradley 

Studebaker Economy Practice Exercises, Scott Foresman 

Maxson Number Cards, J. L. Hammett & Co. 

Courtis Standard Practice Tests, World Book Company 
or any other standard tests 

The Rapid Calculator, Jones 

Compact Efficiency Drills, Iroquois Publishing Co. 

Sample packages of grocery store supplies 

Printed business forms 


Bibliography 

Brown and Coffman. How to Teach Arithmetic. Row Peterson 

Klapper. The Teaching of Arithmetic. Appleton 

Lennes. The Teaching of Arithmetic. Macmillan 

Lindquist. Modern Arithmetic Methods. Scott Foresman 

McLaughlin and Troxell. Number Projects for Beginners. 
Lippincott 

McNair. Methods of Teaching Modern Day Arithmetic. Badger 

Overman. Principles and Methods of Teaching Arithmetic. 
Lyons & Carnahan 

Smith, D. E. The Teaching of Arithmetic. Ginn 

Stone. How to Teach Primary Arithmetic. Sanborn 

The Teaching of Arithmetic. Sanborn 

Suzzallo. The Teaching of Primary Arithmetic. Houghton- 
Mifflin 

Thorndike. The New Methods in Arithmetic. Rand McNally 

Psychology of Arithmetic. Macmillan 

Wilson and Hoke. Howto Measure. Macmillan 


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ef vt. are Te Ode ane 
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ELEMENTARY ARITHMETIC SYLLABUS 13 


by LLABUSMINGARITEMETIC 


FIRST GRADE 


The teacher should read carefully the general introduction to the 
syllabus and should be familiar with the contents of the syllabus for 
the second year. 

Many children enter the first grade with some ability to count 
and make figures. There is need for some number work at this 
time in finding pages in the reader, in saving pennies, in playing 
games. It seems advisable, therefore, to do enough number work 
in the first grade at least to start the child correctly in counting, 
reading and writing numbers within a small compass, thus enabling 
him to use numbers with his games and directing his attention to 
value for his pennies. 

Wrong habits in counting and in making numbers are easily fixed 
at this time unless they are checked by teaching. 

After the first 10 weeks a definite time may be set aside for 
teaching number. This should mean serious work for the teacher 
but play and games for the child. Let the work grow out of the 
natural activities of the school room. 


Condensed Outline 
A Incidental number 
I Counting children, pages, papers etc. 
II Sense training 
II1Il Comparisons 
IV Concept training 


B Formal number 
I Counting — number scale 
1 Counting 1 to 10 
2 Recognizing and reading numbers 1 to 10 
3 Writing numbers 1 to 10 
4 Illustrating numbers 
5 Counting to 100 
6 Reading to 100 
7 Writing to 100 


II Combinations. (For combinations and suggestions as_ to 
teaching read carefully the expanded outline) 
III Measures 


1 Cent, nickel, dime 
2 Pint, quart 


14 THE UNIVERSITY OF THE STATE OF NEW YORK 


Expanded Outline 
A Incidental number 
Incidental number should precede the formal work in 
number and should always be carried along with such 
work. 
I Use all number situations such as: 
1 Counting children present and absent 
Numbering cloakroom hooks 
Keeping calendar record 
Counting out supplies used in class 
Having children number for playing games 
Reading pages of book or counting pages 
Paying for school lunches 
Applying number to handwork 
Counting numbers on clock 
10 Keeping score in games 


LOM COS NISC eC tek CDS 


Il Sense training 
Use games for training the senses of hearing, sight and 
touch. 
IIf Comparisons — Steps preparatory to teaching number 
1 Approximate comparisons 
Several lessons, the principal aim of which is to lead 
children to discern relations between quantities and to 
express them by indefinite terms of comparison, may pre- 
cede the counting lesson with profit. 


The following terms are suggested for comparison: 


larger — smaller 
longer — shorter — taller 
thicker — thinner 
higher — lower 
faster — slower 
wider — narrower 
farther-— nearer 
more — less — fewer 
This approximate comparison may be developed through 
the type lesson or by a game. 
The following game teaches more and less . 


From a pile of pegs the child who is “It” selects any 
number of pegs and the class guesses the number. If a 
child guesses 7 for instance, “It” answers: “I have 


ELEMENTARY ARITHMETIC SYLLABUS 15 


more than 7,” or “less than 7,” as the case may be. As 
each guess is made the child responds with “more” or 
“less.” The one guessing correctly becomes “ It.” 


2 Exact comparison 
For this step, lessons should be given to show that one 
number is more than another namely, 7 is 1 more than 6. 
For this the following game is good 


Bird Catcher 
Select two children as bird catchers, name each row 

after a bird. Mark off one section of the room for a 
cage, another for a bird’s nest. Two bird catchers stand 
in the front of the room. The teacher calls for one row 
of birds to fly to the nest. Bird catchers try to catch 
birds before they enter the nest. The birds who are 
caught must go in the cage. 

Teacher: “ Which has more birds, the cage or bird’s nest? 
Let us count them and see.” Bird catchers form a gate 
and as birds walk through, the children in seats count 
them. 

Teacher: “ There are 10 birds in the cage and 11 birds 
in the nest. How many more birds in the nest than in 
PMescage res 

Children: “ One more bird.” 

Teacher: “Eleven birds are how many more than 10 
birds?” 

Children: “ Eleven birds are 1 more than 10 birds.” 


Application 
Teacher: “Eleven cents are how many more than 10 
Gents tieuctG. 


IV Concept training 
The full meaning of number can not be acquired at 
this time. The following, from Doctor Thorndike, ex- 
presses the situation at this time: 


A number has not one meaning but several. Thus 8 
means a certain point or place in the number series 1, 2, 3, 
4,5, 6, 7, 8, 9, etc., which is 1 beyond 7 and 1 before 9. 
This we call the series meaning. Eight also means the 
number of single size meaning. Eight also means 8 times 
a certain unit, say a pint, whether isolated as 8 separate 
pints or combined together in a gallon. This we may call 


16 THE UNIVERSITY OF THE STATE OF NEW YORK 


the quantity-size or ratio meaning. This ratio pce 
should not be neglected. 

It is not necessary or desirable to teach the full and 
exact meaning of a number all at once, for pupils learn 
more and more about the meaning of numbers by using 
them. 


The good teacher will make sure that, at any stage, the 
pupil knows the meaning well enough to use the numbers 
intelligently in those ways which are necessary but will be 
cautious about teaching any more elaborate meaning than 
that. 

In making a number objective, we should consider not 
only formal systematic presentation but such informal and 
incidental connections as can be made with objects and 
acts of daily life. The latter are more interesting and 
more surely understood. 


Nore. Training in number concept should precede formal num- 
ber and should be carried along for a time with formal number. 
Plenty of material should be available — kindergarten sticks, pegs, 
circles, toothpicks. 

1 Count objects from 1 to 10 or higher 
2 Play and illustrate number by use of pictures and objects 
3 Drill on the numbers as follows: 


Example: To familiarize pupils with number “2” 

Place 2 blue pegs, 2 red pegs, 2 white pegs, 2 green 
pegs, on desk. Drill for other numbers in the same way. 
They may be arranged in geometrical forms as squares, 
triangles etc. or may be arranged to represent houses, 
barns etc. 


The following constructions have been used by the Rochester 
schools for objective representation of number by means of tooth- 
picks, kindergarten sticks of similar materials. 


Jax aX 


One stick boy, His tent, His flower bed, 


ye ti a 
ie hes 


His back yard, | His dog house, His dog, 


ELEMENTARY ARITHMETIC SYLLABUS 17 


Be atc fe ae 
A, es coer 


ha 


His father’s barn, His kite, His back fence, 


His house. 


A good summary at the close of a concept lesson is to have the 
class give all the constructions that they can recall for a certain 
number, the teacher drawing them on the board as each pupil con- 
tributes. Later the figures should be substituted for the construc- 
tions and the work made abstract. 


In this way a rich content may be given to number symbols; eight 
should be more than seven and one; and the child may thus be 
introduced through his own activity to all the fundamental opera- 
tions. It is hoped that the germ thus transplanted may be nourished 
from time to time as occasion makes possible for further objective 
work and rationalization, in order that the child may be gradually 
led to an understanding of all the fundamental concepts and relations 
of number, and become a more independent worker in the upper 
grades. 


B Formal number outlined 
I Counting from 1 to 10 
1 Counting objects 
2 Counting without objects 
3 Recognizing figures to 10 
4 Making figures from 1 to 10 


II Counting to 100 
1 Counting objects 
2 Counting from chart for recognition of symbols 
3 Counting by 10’s using sticks tied in bundles of 10 
4 Counting by 10’s from the chart 
5 Counting from memory 


18 THE UNIVERSITY OF THE STATE OF NEW YORK 


III Written number in connection with the counting 
C Formal number developed 


I To assist the counting have the following chart displayed in 
the room and a small one in the hands of the pupils 


10. 20> 30: 405950" 60% 7/0 Seger 00 ay 
REE PASTRIES seek Rolie copy 22h 
12°22" 32° Ae ees02" fe mee ee oe 
Lo 2333 AB ore OS moon 
14 24 34 44 54 64 74 84 94 
15. 25 435.45 oe 00 3 / oe aeons 
16926 (36:46. 560.06 70° #365896 
17 ©2737 4 RE OLAS, Sar 
IS 2838 °4 Skee OSt 7S ose es 
19°) 29) 39.049 259. 0087S Soleo 


LG) OOS NIC ON Oia Co) Dee aS 
eo: Ga eo Gr 


_—" 


Value of chart 


a Count from chart for recognition of number symbols 
b Count from chart for organization of number in the 
serial relationship so child will see 5 after 4 and 


before 6 


c Chart work makes recall easier 


2 Work to be done from chart and from objects 


Note. It is suggested that kindergarten sticks be used singly 
and in bundles of ten. 


a Count by 1’s to 100 
b Count by 10’s to 100 (row A) 
c Begin at 1 and count by 10’s to 91 (row B) 
d In the same way count each column by 10’s 
e Count by 2’s, emphasizing every second number, as one, 
two, three, four etc. 
f Count backwards 
g Seat work done with chart | 
(1) Use small cards from number boxes 
(2) Have children build the chart with these number 
cards and do various exercises from the chart 
(3) Copying numbers is good seat work at this time but 
should be limited in the amount required and 
should be supervised 


ELEMENTARY ARITHMETIC SYLLABUS 19 


D Formal teaching of combinations 
I ‘Table of combinations 


1 Be ee os aR 7 ed 

Lee eee wie esis le al wl 
2 Violeta thaw 4 im Ldiogl 

CaS Pet a Oe fom OY (reversed) 
5) al thd ni a kl pe 

2 rel ANS Wl Se GSE bes Wile (mixed: 
Abe 3h ald das Sa 

oe ee. (oul eS } 
Optional 
5 De Ame MG S/s5 co 

en oon 2. AOA e in CoMmbinattol) 
Fer MNS. EB MSitiae. Ssccieds 

4 5 6 /7 (3 in combination) 
4s 4 4 

5 6 (4 in combination) 


Rows 5, 6 and 7 may be deferred until the second 
gerade at the option of the teacher. But if the child has 
use for these number facts, it is recommended that he 
be given the opportunity to acquire them. 


II Teaching 

1 Develop combinations through pictures, objects, domino 
cards etc. 

2 Give number symbols 

3 Make the combination a matter of memory. Transfer 
combination to flash cards and drill. Do not allow child 
to keep resorting to objects until he depends upon them 
for counting. See that he depends upon his memory. 

4 Problems for application as: Mary had 5 pennies and her 
father gave her 3 more, how many pennies had she? 

5 Standards of speed and accuracy: Ten known combina- 
tions in about 15 seconds given orally from charts, lists 
on board etc. is a good standard. 


Optional 
1 The more difficult combinations as given in the table. 


20 THE UNIVERSITY OF THE STATE OF NEW YORK 


2 The concept of subtraction 8 less 4, or of multiplication 
2 times 4, or of division 2 fours in 8, or of % of 8 is 4 
may be developed in the constructive work and expressed 
orally. 

It is recommended, however, that symbols expressing 
such ideas be deferred until second and third grades. 


First Grade Equipment 


Kindergarten pegs and sticks 

Rulers with inch divisions 

Yard stick 

Addition cards with combinations and answers on the back 

Domino cards with combination on one side 

Toy money 

Abacus 

Ringtoss 

Ring a peg 

Ten pins 

Bean bag 

For illustrating the number stories the domino cards are good. 

Have the domino illustration on one side, the number symbols on 
the other. 


been 0 
| @) 5 
O ) 
O 
O 3 
O 


These can be used for giving the number idea, the symbol and 
games. For drill use the figures only. 


Devices and Games for Drills 
1 Place flash cards along blackboard. Children run, take card 
and give combination. When all cards are chosen, children hold up 
cards at seats. One child takes place in front of room and says: 
“Bring me 2 and 3 are 5.” Then child hands the card to the leader 
saying: “I bring you 2 and 3 are 5.” Continue until all cards 
are gone. 


ELEMENTARY ARITHMETIC SYLLABUS Za) 


2 Divide class into boys and girls. Leader calls for 2+ 3 are 5. 
A boy and a girl run up and hunt for 2-+3=5. First one may 
have the card. Keep score for boys and girls. If each score gives 2, 
this will give a good application for counting by 2’s, or counting 
by 10’s or 5’s may be used in same way. 

3 Teacher: “I am thinking of a combination on the board” (or 

cards). 
Children: “ Are you thinking of 3 and 2 are 5?” 
Teacher: “ No, I am not thinking of 3 and 2 are 5.” 
Continue until a child gives the right combination. 
Teacher: “ Yes, I am thinking of 3 and 2 are 5.” 


4 Draw lines on board to represent stalls. In each stall place a 
figure as 8, 5, 6, 4 etc. | 

Teacher passes out cards. Children run up and place cards on 
the chalk tray below the stall whose number equals sum of com- 
bination on card. 


5 Ladder drill. Combinations written on a ladder, children climb 
ladder by giving combinations. In a similar way use fish in a pond 
and allow children to go fishing. Use fruit trees and allow child 
to pick fruit. Use brick wall with combinations on bricks. Children 
tear down wall. 

6 Scoring games: (a) bean bag, (b) ringtoss, (c) nine pins. 

7 Racing game. Place ten or more combinations low down on 
different parts of the blackboard. Let a boy and a girl be chosen 
to write the answers, each beginning at the word “go” to find 
which can write the answers the more quickly. If there is sufficient 
blackboard more than two may race. The winner chooses some 
other pupil to race against the loser unless the teacher desires to 
substitute another child. 


Store Projects 


1 Material: Kindergarten table for counter, objects to sell made by 
children, money 1, 5, 10-cent pieces made by children, change 
drawer. The articles may include apples, pears, dolls, furniture, 
plasticene dishes, etc. 

2 Grading difficulties as follows: 

a Counting to 10 cents, 1-5—10 
(1) Purchasing single article, exact amount of money, but less 
than 10 pennies 
(2) Purchasing two articles using count — 10 pennies 
(3) Purchasing one article with change, using 5-cent piece 


22 THE UNIVERSITY OF THE STATE OF NEW YORK 


(4) Purchasing one article with change, using 10-cent piece 
(5) Purchasing two articles with change, using 10-cent piece 


3 Procedure 
The above steps are taken up in the following manner. 
This may be carried on any time in the year adapting the 
difficulties to the number knowledge which the children have 
before progressing to the next. 
a Arrange the articles for sale on a kindergarten table, each 
bearing the cost mark from 1 to 10 cents. Each child is given 
10 pennies with the privilege of buying one article. Teacher 
selects a child to go to the store. 

Customer: “ Good morning?” 

Storekeeper: ‘Good morning.” 

Customer: “I will take a pear, 2 cents.” 

The customer counts out 2 pennies, hands them to the store- 
keeper who deposits them in the penny compartment of the 
money drawer. Continue as above. 

b For preparation, review combinations. First using the different 
serles as: 


D7 PG2T P2EIWS Su 0 
lieei2 (d3° well eee SS rete 


When children grasp this step, skip around using combination 
in different series. Use the same arrangement of the store as 
given under a. Each child is given 10 pennies with the 
privilege of buying two articles. Cost of articles 1 to 9 cents. 
Teacher selects child to go to the store. 

Customer: “ Good morning.” 

Storekeeper: “Good morning.” 

Customer: “Iwill take a peat,@2 .cents, andsa doll Socents, 
2 andsnpares/. 5 

Customer gives the storekeeper 7 pennies. 

Storekeeper: “ The pear: costs 2 cents and the doll 5 cents, 
2 andsocarer7. No muauces: 

c Arrange store as suggested under a. Each child is given a 
5-cent piece with the privilege of buying one article. Articles 
are marked 1 to 4 cents. 


ELEMENTARY ARITHMETIC SYLLABUS we) 


Preliminary step: Review combinations at board as: 


(1) beeZ 
4 osy3 
es 
(2) ee Children run up and fill in the 
ee missing number. 
Saison 
(3) Lying 
fers 
aes 


9 


Customer: “ Good morning. 

Storekeeper: “ Good morning.” 

Customer: “I will take this chair, 3 cents.” 

storekeeper :. > Lhe chair costs:3 cents,..3.and.2.are.5..+2 tents 

change.” 

d Proceed with this step as under c. A greater range of com- 
binations will come in. 

e Arrange store as suggested under a, each article marked from 
1 to 9 cents. Each child is given a 10-cent piece with the 
privilege of buying two articles with change. 

Customer: “Good morning.” 

Storekeeper: “ Good morning.” 

Customer: “I will take this fish, 3 cents, and this cup, 6 cents. 

3and6are9. Qand1lare 10. 1 cent change.” 
4 Other applications 7 

a Suggestions: Christmas store, Christmas tree, party, picnic 
food, children’s lunches when served in school 

b Type problems of maximum degree of difficulty 

(1) Robert had 10 cents and bought at a toy store a top for 
3 cents and a boat for 2 cents. How much change did 
he receive? 

(2) Mary’s mother gave her 5 cents. She bought some candy 
for 3 cents. How many pennies did she have left? 

(3) Arthur spent 5 cents for a kite and 2 cents for a ball. He 
gave the storekeeper 10 cents. How much change did 
he receive? | 

(4) Mary picked 4 apples and Frank picked 3 apples. How 
many more apples did Mary pick than Frank? 

Encourage the children to suggest original problems as far as 
possible. 


24 THE UNIVERSITY OF THE STATE OF NEW YORK 


SECOND GRADE 
The teacher should read carefully the general introduction to the 
syllabus and should be familiar with the contents of We syllabus 
for the first and third years. 


Condensed Outline 


Note. The following outline and treatment are not to be interpreted by the 
teacher as a requirement. She should feel free to use other arrangement or 
treatment of topics if she believes it advisable. This plan of work has proved 
successful. 


1 Reteach first grade number work 


a Counting, reading, writing numbers to 100 
Use number chart to establish a good understanding of the 
number scale. Watch for the children who have immature 
number ideas. There is often such a group in the second grade. 
Use objects, pictures, domino cards for these children to illus- 
trate number but drop the objective work as soon as possible. 
b First twenty-five combinations taken up in first grade 
Here again use object illustrations, until the number idea is 
clear and then omit such work. See first grade outline for 
development lesson on 8. 
c Money taught in first grade, 1¢, 5¢, 10¢ 
d Problems by teacher on the above 
Original problems by children 
2 Addition without carrying 
For teachers’ reference the combinations are classified in the 
following groups: 
a Combinations whose sum is 10 (group A, see page 27) 
_ 3 Addition carrying 
4 Subtraction 
Use the ten combinations for drill on the addition form of 
subtraction thus 10, saying 5 and 5 are 10 (see page 30) 
—5 
5 Addition continued 
Take up all the doubles (group B) one set at a time and give 
all drills as outlined (see page 32) 
6 Multiplication j 
Table of 5’s (see page 28). Use in examples and applied 
problems 
7 Addition (group C) using 9 endings with series and column drills 
(see page 34) 


ELEMENTARY ARITHMETIC SYLLABUS 25 


8 Subtraction using 9 series with examples for continued drill and 
addition span (see page 36) 
9 Multiplication 
Table of 2’s with examples and problems. Use same method 
as in table of 5’s 
It is suggested that the 2B work may be ended at this point. 
10 Reading and writing to 1000 
For groups D to K see pages 34-35 | 
11 Addition (group D) using 8 endings with series drills, column 
addition and problems 
12 Subtraction with changing minuend and subtrahend 
13 Table of 10’s 
14 Addition (group £) using 7 endings, with series, column addi- 
tion and problems 
This completes the groups of combinations used in series and 
column work. The remaining combinations are learned but not 
used in column addition until the 3d grade. 
15 Subtraction completed — all groups 
16 Addition (group F) using 6 endings without series drill or 
column addition 
Group G 5 endings, group H 4 endings, group L 3 endings, 
group J 2 endings, group K 1 ending, thus completing all com- 
binations 
17 Subtraction continued involving all combinations 
18 Multiplication tables of 2’s, 3’s, 4’s, 10’s 
19 Fractions %, 1%, % of products learned in multiplication tables 
are optional 


Hints to the Teacher on Addition 


Addition can and should be acquired early. See that the child _ 
does not go through school poorly prepared, counting on his fingers, 
a slow process. Help him to be quick and accurate. He will then 
have a foundation for later work. 


Essentials 
Have a well-graded system of presenting the difficulties in addi- 
tion, and present a few at a time. 
Difficulties in Addition 
1 Gaining a knowledge of the combinations 


26 THE UNIVERSITY OF THE STATE OF NEW YORK 


bo 


SOR COO’ wm 


NO 


Applying the combinations to the higher decades or the series 


WOTk,.aS 2/1 4/ wos, 
Os Le BO 


Note. For the most gifted child the knowledge of 8+7 <= 15 prob- 
ably never insures the application to 38 + 7 = 45. 
For the less gifted child gaining facility in the decades involves much 


time and labor. : 


Learning to keep in mind the unseen result of each addition as 


involved, namely, column addition 

Adding this unseen result to the seen number 

Acquiring power to hold attention when any considerable number 
of digits in a column is involved 

Learning to skip 0’s 

Learning to skip empty spaces 

Keeping one’s place in the column 

Carrying 


Methods of Meeting the Difficulties 
Have system well graded as to difficulties 


Master a few combinations at a time 

a See that children early understand the combinations by use of 
objects 

b See that the use of objects is later discontinued and that children 
memorize the combinations 

c Look out for counting. Do not allow a child to stop for 
counting. If he hesitates, he does not know his combination 
result. Better give him the answer and drill upon it. 


As soon as a few combinations are known, introduce the work on 
the decades 

a After the first twenty-five combinations (see Ist grade) are 
known the work on the decades should begin 


b Make each series a matter of memory 


As soon as the combinations and the series are well started give 
plenty of practice on column addition. Column should con- 


tain only a small group of the combinations. 
a Have plenty of practice on the column. Let the child add aloud 


some column over and over if need be until he knows what is 
expected of him in smoothness and speed. 

b Addition is memory work. By the time he has memorized many 
addition lessons on columns he will have covered many dif- 
ferent applications of the same combinations. 


ELEMENTARY ARITHMETIC SYLLABUS 27 


5 Advance pupils into new work very slowly so that each step may 
be done thoroughly 


6 Teachers should keep a record book of all the combinations taught 

and the problems and examples containing these combinations. 

This will mean a saving of time as plenty of material will be 
available for drill. 

Courtis says: “ By practice on one example until it is learned by 
heart convince the child that it is quicker and easier to remember 
than to count.” 

7 Children may know combinations and series and still fail on 
column addition. The difficulty is in the child’s power to hold 
the unseen number in mind and unite it with the next figure 
in the column. . 

Courtis says: “ Most children add steadily for 6 or 8 addi- 
tions. Some have an attention span of only 3 or 4 additions.” 

The remedy is practice. Practice only will train the attention 
and make column addition possible. 


8 Carrying 
Teach the child to carry the figure immediately and not hold 
it in suspension in the mind until the end of the column. 


Detailed Procedure for the Second Year 

For each fundamental operation have some plan well-graded as 
to difficulties. Several plans have recently been made available in 
various courses of study. The following plan arranged after the 
plan in the city of Toronto Exercise Book is given here because it 
has proven very successful. The advantages are: 
1 More easily memorized combinations are mastered first 
2 A power to hold number in mind is given through column addition 

while the child is still working with the easy combinations 

3 It grades the difficulties for subtraction 


Group A 


Deer Oe poe) hie Pmt ao a eo. eA 5 
Slog he aie ae Rees 288 Eu Se el Area ot Ces 


Step 1 Review the above group. Arrange class into groups for 
drill. For those pupils that need objects resort may be 
made to some such work as is given for the first grade 
on teaching number concept. Follow with plenty of 
abstract drill. 


28 


THE UNIVERSITY OF THE STATE OF NEW YORK 


Step 2 Series work with each pair of combinations taking only one 


set at a time thus: 


a 55925 BS 
5.4)d0: Saya 
10 20 30 40 etc. (with answers) 
b 54 fly doo 
Dubbed {yi eed uO 
C cw ore etc. (without answers) 


10 20 30 40 etc. (without second addend) 
Rapid drill on the series may be carried to 100. 


Recite as a table. This is also good drill. Teach at this 
time also the tens series with any digit thus: 


Des tele OeiiaO itt commie 
TO (20%, 30 LOL Ome ZO sera 


If a teacher desires, this lesson may be given with bundles 
of 10 sticks and single sticks thus: 

Take a 10 bundle and 4 pegs. Child says 10+ 4= 14. 

Take a 10 bundle and 6 pegs. Child says 10 + 6= 16. 

Take two 10 bundles and 7 pegs. Child says 20 + 7 = 27. 


Step 3 Column work 


Column of figures should be planned so that each column 
contains only known combinations and known series. Many 
columns should be given so that each step is completely 
covered by column work. Do much oral work. 

Require smoothness and accuracy in recitation. If a child 
has difficulty, he may be encouraged to practise on his 
column. 

Add columns from bottom to top. Do not allow grouping 
at this time. 

Purpose here is to give drill on these separate com- 
binations. 


owmunuiNy 


Y 
SEs) 
ee a 


| cnt U1 U1 U1 OD 
| orn U1 1 U1 Ut 
| WUT ON 1 


Written work may be given. 


ELEMENTARY ARITHMETIC SYLLABUS 29 


Step 4 Problems apply to above as counting nickels. Original prob- 
lems by children. 


Examples of original problems given by children. 


If I had 10 cents and spent 5 cents for a bag of peanuts 
and 2 cents for an eraser, how much change do I get back? 


If I buy a 5-cent tablet and three sticks of gum at a penny 
a piece and a 2-cent sucker, how much do I spend? 


Series Drills 


Take up each combination in group A and work through all the 
above steps. The following is work for 1+ 9 of Group A: 


Step 1 1+ 9,9-+-1; call attention to 0 ending. 
Step 2 Series l 
9 


11 
2 
20 30 50 40 (with answers) 
11 
0 


| Hos 


(without answers ) 


Deb ah ae ee 


10 20 50 30 (without second addend) 
Step 3 Columns involving 9+ 1,1+9,5-+5 


mV moO em OS 


= Ore OO un 
mine Ore OO UwM\o 


mmr OWN 
wm Ute OO OV 

Ore uu OO W 
mmnNorreuut 


Make up many columns similar to above. 
Step 4 Problems, games for scoring, races. 


2+ 8 Series 
Step 1 2+ 8,8+ 2; call attention to 0 ending. 


30 THE UNIVERSITY OF THE STATE OF NEW YORK 


L222 SAP ase 


Re] S| ie Ye 
seas shales hal ate: 


(pide Sein Neh dem 3): 
abaiel sie he 


2 
8 
10 20 30 50 40 (with answers) 
8 
2 


(without answers) 


Bre aes 


10 20 30 40 (without second addend) 


Subtraction 10 10 10 10 10 
Se ee ee eS 


Step 3 Column work for 8+ 2,2+8,9+1,1+9,5-+5 


6 7 
Deel Z 5 
A Sin eos) SHOE, DIE 25 
Dit Duar eae ROBE eres 
Si dep ream nek eee ke 
Be we) Pee eee. ae Dee 
oo le Poems Beye 
SEL EES 82 oe 
De Wah Di Rie ie a et oes 
Step 4 Problems, games etc. 
3+7 Series 
Step 1 7+ 3, 3-+7; teach O ending. 
Steps Fav 2 / TU os: 
ON 3 oa ees 
10 30 20 40 (with answers) 
Stepao 3 gcd ees 
{32/6 33/4 
(without answers) 
Tee /olaed, 
10 20 30 (without second addend) 
Subtraction 10 abel Oe LO amen 


—7 —8 —9 —1 —3 —2 


ae 


ELEMENTARY ARITHMETIC SYLLABUS 31 


Column work for 7+ 3,3+7,8+2,9+1,5-+5 


7 
Gm 638 406 
LP io raat iin $3 habit obec 
cy Wig Ad LE ZEN GN 2 
Joe Soke oe Pk 
SAT 7ey ae oe Gee s5 
Vee pas) \ td's of Bas) to kelp t Va, 
3. fee 0 ot /e ee Le atc! 
Step 4 Application 
4+ 6 Series 
Step 1 6+ 4, 4-6; teach 0 ending. 
GO lhe rea0 
4 4 4 4 
10 20 30 40 (with answers) 
Step 2 Ge lUSs 207/30 
4 4 4 4 
(without answers ) 
Oe LOmeZO 46 


10 20 30 50 (without second addend) 
Column work for 6+ 4,4+-6,7+3,8+2,9+1,5-+5 


Se 

Sea 3 PAG 
5 Se. eed re Gee Cs forest 
Ae Oe ree, 80) AS 
ee tS ie Peete Gy ean ee 
Ge 4 on) Ce ere tL 
47, (GMO bre 4D 2" 9 
Ger Age. 9 eee: OR2 ESS 
HP Cy Loon) ae Fake fete 

Hints 


1 Give column cf 4, 6, 8, 10 addends. 
2 Call lower figure 24 or 34 or 14 to throw the drill into 
the upper decades. 


Step 4 Problems etc. 
This completes work on group A. 
Take up each group in the same way. 


a4 THE UNIVERSITY OF THE STATE OF NEW YORK 


Carrying 
At this point begin double columns and teach carrying. 
After carrying is established, written work may be given. 
DO Joe 7 PiOAn 7 
Shh RDN ESS 


JO Le OomeLo MLS 
AD i Ome Be tO 


eee 


In making up double or triple columns take care to carry a figure 
to the next column which makes some known combination. These 
examples have been arranged for addition to begin at the bottom. 


Zh eB 
/Spig OS hie 
37: A/a 136 
43 53 64 


Group B 


Take up each group in the following way. See Toronto Exercise 
Book. 


Ui a Ean) Ud Sra et il ane 
ONO 2/2 0 AG ee ea 
Step 1 9+9 
Step. 2 Series iy 7 aes Olga sD 
DG BAO TESG 19 
2 oa 
OM eA mee i 2 
Of aD Oe ewe tO 
2 tale oe eee Pal Cote eG) 
SPREE oh ERD OE ec at 
DELS OR BOO fas aia 
PAE AU GS Tid Comme nS 88 
Dad sa CO 9 es eran 
Oe Os ek eee ae 


ELEMENTARY ARITHMETIC SYLLABUS 


Children will feel this new difficulty for a short time. 


53 


If they 


have much trouble on the 10’s, it will help very much to write the 
sums out at the side and allow a little time to child for study, thus: 


9 
2 40 
ac eso Go 
9 29 29 459 
2 92 929 
9 GOF US2 
9 79 778 
8 + 8 Series 
Step 1 8+8 
mBtepe2 ~eries 8 18 28 48 38 
cy Roy c kee eases 
Step 3 Columns | 
5 
Oe Pat cee 
hg Ae eases anne? ed’ 
DRAG OG 94s). 88 1O0 
peo. 9 BO Rae As 
eer aR Oa Se 
Seo? LOmAIG ONT teRED A So 
Bie) Cua. aa 0 tk Oo 
Step 4 Problems 
7+7 Series 
Step 1 7+7 
Step 2 Ti OF PSS fey 
DAV ERP TA EENY, 
Step 3 Columns 
nO), 
oe 0 Deere AZ6 
OT IN ae eee 9G 
Sees ee eg moro, 
Oh” Ofte tare ee et O0G/ 
Gye Ge MA GPAs 77/6 
Atel See pee if 7 
(hal Aatoe ESt Df, 


546 


. 34 THE UNIVERSITY OF THE STATE OF NEW YORK 


Step 4 Problems. Complete 6 + 6 

(1) Through the series 

(2) Through single columns 

(3) Through double and triple columns 
Complete 4 + 4 

(1) Through the series 

(2) Through single columns 

(3) Through double and triple columns 
Complete 3 -+ 3 

(1) Through the series 

(2) Through single columns 

(3) Through double and triple columns 
Complete 2 + 2 

(1) Through the series 

(2) Through single, double and triple columns 


Complete 1 + 1 
(1) Through the series 
(2) Through single, double and triple columns 


Review of above 


8 (0) 20.) 07 la See eendinewisee 


Review all other endings. 


Take up group C with each of the steps covered in Group A. 


Group D 


SHG Hil, Bom 
BE Aya Te Tate May isictabhateg shore’ 


Cee 


ELEMENTARY ARITHMETIC SYLLABUS 35 


Group E 


Gimbare (hilt OadViapsine 2 hn 
le Boils Ze DSi ltOan Al vidn aOvendingist/ 


It is recommended that the second grade does not carry the series 
and column addition beyond this group. The following combinations 
should be taught thoroughly but it is recommended that they be not 
used in series and column addition: 


Group F 
Peek? & ©. ORR SO ee 
So 7 lS 2. Sgennineiss 
Group G 
Ie San Zvi 4 AS nye OD 
Ape cee So hOD le Zemie -Ovendifietis © 
Uy ae 04 ah aus 4 apa Pt A ibe 
Group H 
|RSS eet ss kee ee = 
Seno nah eos  O<eticing is: 4 
Group I 
heer Ge ean @ at OO oo. 
Beater, eee ol eee oe Dr endingoisns 
Group J 
Seow at a7, a UNS 
OT i Che ee ee ree els. 2 
Group K 


ATS BOBS AG 
APPA Pag otc) ending Tis lL 


= 


\O BO 
COW 


Standards: Answers to 10 combinations in 12 seconds. 


Subtraction should be carried along with addition. Use the 
Austrian Method as this makes use of the combinations already 
learned and is of immediate use in making change. 

For a clear exposition of subtraction with its disputed points read 
Doctor Thorndike on subtraction in his New Methods in Arithmetic. 


36 THE UNIVERSITY OF .THE STATE OF NEW YORK 


Use the “and” form of getting the difference and leave the idea 
of take away or minus and remainder for third grade work. These 
terms and their meanings may be developed in third grade after the 
process of subtraction is well established. 


1 Ten endings 


a After tens combinations are learned, drill thus: 


SP ey MRE Ne 
i Vite Caaae 


10.10. 10° 10 ete) sayine-9 and.) make: 10: 
Use varied arrangements 
b Use this form: 
10% SLO Omer 
—9 —/7 —6 —4 saying 9 and .... make 10. 


c Apply in problems making change for 10 cents 
2 Doubles 


18) 16°84 aS a Ge epee 
—9 {2B 7 eae ee 


3 Nine endings. 


a. 90 50 1810s Cin nO me se 
man | eld 2G, ee ee ee eg) ees 


a 


b Continued subtraction but no carrying 


99 999 999 999 
76) O10 oom 


Give plenty of work similar to this. 


c Introduce 10 in the minuend, as 


109) 2 4109 Get O9 Rog 
—8/ —64 —25 —90 etc. 
Seven and what are 9? 
Hight and what are 10? 


d Continued work for span of attention 


1099 10999 
—463 —8642 etc. 


ELEMENTARY ARITHMETIC SYLLABUS ad 


4 Eight endings 


a Se ke! ey eee a eee S 
AT Sag ec! pal ds taeda ay naa a) 
b 888 888 888 8989 
AFA 904 73 14 tet: 


c Introduce carrying or changing both minuend and subtrahend 
at this time, thus: 


80 change 0 to 10 5 and 5 are 10 
—25 change 2 to 3 3 and 5 are 8 


os 


Simpler and fewer words used here the better. 


d 80 80 
—36 —45 Much drill on each step. 


e 9980 9890 
—7246 —4319 One change here. 


he 1860 1890 

—946 —958 Two changes here. 
g 880 

—285 Continued changing. 


5 Seven endings with the various drills 
6 Continue throughout the successive groups as in addition 
a Problems should be given each day applying the number drill 
b Making change 
For this use store project. A few suggestions are given but 
many adaptations are possible. 
(1) Teach coins 1 cent, 5 cents, 10 cents, 25 cents 
(2) Make change from 5 cents 
(3) Make change from 10 cents 
(4) Make change from a dime and a nickel 
(5) Make change from two dimes 
(6) Make change from 25 cents 


Suggestions 
1 Have a box of toy money for each child. 


2 Make charts with pictures of toys and price attached or make 
real articles in class and use in store. 


38 THE UNIVERSITY OF THE STATE OF NEW YORK 


3 Teacher states problem and later children will be able to state 
problems. 

4 Children altogether, one child aloud makes change thus: The 
toys cost 7 cents, counts 8 cents, 9 cents, 10 cents putting down 
the pennies and saying “ My change is 3 cents.” 

5 Later children make change without counting aloud, just giving 
the amount. 

Multiplication 
1 A good order for taking up multiplication tables in second grade 
is:5'S:'2'S.cL0 St Gua omaus) 
2 Introduce the operation of multiplication by saying from the 


illustration: 
2 fives 10 5 5 5 =) 5 
5 5 5 5 5 
3 fives 15 a 5 5 5 5 
4 fives 20 ee 5 5 5 
5 fives 25 ie 5 5 
6 fives 30 FEES 


3 Build up in table form but do not have it memorized in tabular 
form, as the child may then resort to saying the whole table to 
get one fact from it. 

4 Have facts memorized in broken arrangement. 

5 Use flash cards, games, races and other devices for drill. 

6 Application 

a As soon as a few facts from the tables are learned, use thus: 


32) 09 640 
X20 Kee 


Se eannmeeeenttt T 


b Apply te eddy as follows and at this time introduce the 
expression “ times.” 


It costs 5 cents for 1 bag of marbles, it will cost 2 times 
5 cents for 2 bags of marbles. 


It costs 5 cents for 1 bag of marbles, it will cost 3 times 
5 cents for 3 bags, etc. 

Toys that cost 4 nickels cost 4 & 5 cents or 20 cents. 

Toys that cost 5 nickels cost 5 & 5 cents or 25 cents. 

Toys that cost 3 nickels cost 3 & 5 cents or 15 cents. 


Type of analysis for young child. 


One stamp costs 2 cents, 5 stamps cost 5 X 2 cents or 10 
cents. They cost 10 cents. 


ELEMENTARY ARITHMETIC SYLLABUS 39 


Arithmetical Situations 


Two situations present in the schoolroom are given here with 
suggestions for their possible development. They can be used to 
create a need for number, as an incentive for gathering information 
and as an incentive for drilling on number facts to develop skill. 

The first situation consists of the supplying of milk to the children 
in the second grade room. This has been a source of real arithmetic. 

The second is a project undertaken by the class, the play store. 
This is a common and popular method of making arithmetic real 
to children. The store project permits of adaptation to all primary 
grades giving a great variety of arithmetical information and 
experience. . 

Project. Create the need for and make a practical application of 
number. 


Situation. Milk furnished by the school and bought by the pupils in 
the schoolroom proved to be a fruitful source of arithmetic for 
the term. Each day children brought their money and records 
were kept of the cost of the milk and amount of milk used by 
class per day and week. 

Teacher's aim 


1 To count by 1’s 

2 To count by 2’s 

3 To teach dozen and % dozen 

4 To teach money value and make change for 5 cents 

5 To count by 3’s the amount of money brought for each child 
for the week 

6 To teach 2 pints make 1 quart 

7 To teach ¥% as applied to pints 

8 To teach child to read cents 

9 To teach child to read dollars and cents 


Pupils aim. To buy milk and make change 


Development 
1 To count by 1’s. Each bottle contained % pint. Use term 
YZ pint. 10 
Count number of bottles and keep a record each 11 
day for a week. This gives an idea of the use of f 
column addition. 11 


2 To count by 2’s 
Counting by 2’s the % pints to get the number of pints 
used each day. Keep a record. 


40 THE UNIVERSITY OF THE STATE OF NEW YORK 


3 Count number of wafers and straws 
Use term dozen and half dozen 
4 At 3 cents a piece children make change for 5 cents and 10 
cents. Also teach money value. Children count money in 


the box. 
5 cents = 1 nickel 
10 cents =1dime 
2 nickels = 1 dime 
25 cents = 1 quarter of a dollar 


Project. The grocery store suggested by the children during conver- 
sation with the teacher. 


Teacher’s aim. To create an interest in and need for number, for 
knowledge of the number combinations, for skill in their use, 
ability to make change, some knowledge of measures, and infor- 
mation relative to food values. 


Pupils’ aim. To play store. 
Preparation. To make the store. 


1 Children obtain a dry goods box from a store. Children 
measure box with ruler and learn height and width of box 
in yards, feet, inches and half inches. Line the box with 
manila paper to fit box. Let children decide distance between 
shelves. Make shelves of thin pieces of wood. Use small 
table for counter. 


2 Make money from oak tag tracing around real coin and marking 
the value. Ask children what coins will be needed in store. 
How many pennies in one nickel? 
How many pennies and nickels in one dime? 
How many nickels and dimes in one quarter? 
How many quarters in one 50 cents? 
How many nickels and dimes in one 50 cents? 
How many half dollars, quarters and dimes in $1? 


3 Children bring clean empty boxes and cans from home and 
arrange neatly on shelves. 


4 Make price tags. This necessitates the children visiting the © 
store and finding the actual cost of articles sold in the store. 


5 Children choose name for store. 


Playing Store 
Children choose cashier and storekeeper. Teacher may take part 
of mother until children can do it. At first children buy only one 


ELEMENTARY ARITHMETIC SYLLABUS 4] 


article costing less than 10 cents. This will give them the idea of 
subtraction as used in making change. 

The following form is suggested : 

Mother: “ Here is 10 cents. You may go to the store and buy 
one box of matches.” 

Child goes to store. 

Customer: “ Good morning, Mr S.” 

Storekeeper: “Good morning, Mary. What would you like this 


morning?” 
Customer: “I would like one box of matches.” 
Storekeeper: “It will cost you 6 cents.” 


Storekeeper writes price on small slip of paper and hands to 
customer. 

Storekeeper: ‘“ Pay the cashier.” 

Cashier: Looks at slip and takes 4 cents from box and says, “7, 
8,9, 10. Your change is 4 cents.” 

As children learn more combinations and acquire ability to buy and 
make change, let them purchase two articles and then several. 

When child buys two of the same article he will see the need of 
learning the table of 2’s and so on with the other tables. When 
buying two or more articles children will see the necessity of learning 
combinations well and of column addition. 

Customers should see that storekeeper and cashier make no 
mistakes. 

The store idea may be varied to suit age and ability of class or 
may be adapted for higher grade work. ‘Thus the toy store, or five 
and ten-cent store is good for making change from 5 and 10 cents. 
Bakery shop may be used to teach for dozen and half dozen. The 
grocery store may be used to teach the use of measures as pint, 
quart, gallon, peck, bushel. The butcher shop may be used to teach 
pound, ounce, fractions. 


Make change in the following steps. 
1 From 5 and 10 cents 
2 From 25 cents 
3 From 50 cents 
4 From 1 dollar 


nu) THE UNIVERSITY OF THE STATE OF NEW YORK 


THIRD GRADE 
The teacher should read carefully the general introduction to the 


syllabus and should be familiar with the contents of the syllabus 
for the second and fourth years. 


Ny" ©), Os Cos tS) 


24 


20 


Condensed Outline 
Addition 
a Review all combinations a few at a time 
b For column addition review group 4, ending in 10 
c Review second grade outline group B, doubles 
d Review group C, ending in 9 
Subtraction: groups A, B and C 
Multiplication tables of 5’s, 10’s, 2’s and 4’s 
Multiplication by above multipliers 
Measures: United States money 
Problems 


Addition: All combinations with stress on group D, ending in 8, 


group E£, ending in 7, group F, ending in 6 

Subtraction: Same groups as addition 

Multiplication tables of 3’s and 6’s 

Division tables of 5’s, 10’s, 2’s and 4’s 

Fraction form of table of 2’s: % of 2 etc. 

Measures: linear, liquid, dry, weight, United States money as a 
need is felt | 

Problems 

Addition. Review combinations, group G, Ses. in 5, group H, 
ending in 4, group I, ending in 3 

Subtraction: Groups as for addition; complete all subtraction 

Multiplication tables of 7’s and 8’s with review 

Division tables of 3’s and 6’s 

Fractions: 1/7 of, 1/8 of; part taking tables 

Measures 

Problems 

It is suggested that the work of the first half of the grade 

end here. 

Addition: group J, ending in 2, group K, ending in 1; review of 
all addition. 

Subtraction completed and reviewed 

Multiplication tables of 9’s, 11’s and 12’s. Tables of 11’s and 
12’s are optional 

Division tables of 7’s, 9’s, 11’s and 12’s._ Tables of 11’s and 12’s 
are optional 

Division process 


| 


| 


| 
| 
| 


ELEMENTARY ARITHMETIC SYLLABUS 43 


26 Fraction form of tables 
27 Measures 
28 Problems 
29 Addition with oral and written problems 
30 Subtraction with oral and written problems 
31 Multiplication with oral and written problems 
32 Division with oral and written problems 
_ 33 Reading and writing of numbers 
34 Denominate numbers: linear, liquid, dry, United States money, 
weights 
35 Proofs 
Full vocabulary for each process acquired. 
Tests: Courtis Standard Tests. Class should be up to the general 
standard of the Courtis test. 


Expanded Outline 


1 Notation and numeration 
a Counting, reading, writing numbers through three, four, and five 
places as need for them is felt 
b Roman numerals to XX; apply in reading the clock face and 
chapters of books 


2 Addition | 
Reteach work of second grade (see second grade outline for 
addition) 
Group A 
hs AS Lee Pe hy ead « ee Aerts 3 ae 
5 1 2 3 4 andreverses 9 8 7 _ 6 ending is 0 
Group B 
ST gah AT ailetioden wre) LSP OMA Tg 
pee u/s | Oe eee 2 
Group C 
Om BAIA cA oe (ON (0), oe aie 
WO e/a tote i ending, 1g: 9 
Group D 
Varo. SOU EE. aioe 
Lae o Ase. |G endingsis"s 


44 THE UNIVERSITY OF THE STATE'‘OF NEW YORK 


Group E 


bie Oe? A Om ier aan 
2° RE Ge ea oe 2 rent, 


Suggestions 
1 Teach group and the reverses. 
2 Give rapid drill on the series table. 
3 Column addition 


a Planned so carefully that only the combinations of the immediate 
group or review group are met in the columns. 


b Have single columns of 4, 6, 8, 10 addends. Have double 
columns with carrying. See that the figure carried makes 
known combination in the second column. Always have figure 
carried in the beginning of the column and plan to add from 
the bottom to the top of column, otherwise the scheme of 
grading the difficulties will be lost. 

c Have examples of three columns. 

d Have problems by teacher. 

Have problems by pupils. 
e Have game for scoring races etc. for drills. 


f Teach children to read the result saying the sum is ... 


Group F 
Group F starts the new work of the third grade. 


step “Lin nl ea ee ae 
511 Ae ee Le a 


Beotenee 
a 2.7 22)" \Zaeet eee 
4 Ages aaa 
6) .-26. 167,46 36 
b Zwe2e el 20 epee 
4 4 4 4 4 
C 


ELEMENTARY ARITHMETIC SYLLABUS 45 


Step 3 (4+ 2) Ze (7+ 9) (1+ 5) 

2 ee) Gor Os oe hig 

Aiea eZ. WAG) BOs Mad and oniromet Y 

eee One 7 eA ORM 1), Be c/n O62 

meee 9 tes eae Or. Pana Ee B8O 

ABAD LOMLOLN! OA RAL NOG AID 93902 

ee po On de LZ" Se Ae eee ead 

Bee Ae A 8/9 AD Bae Sa ee 27, 

ewer rt ROL) Mae oe) eens OOD 

1414 9497 


4432 4669 5672 


Step 4 Problems 
Teachers should make up plenty of columns similar to the above 
in step 3. Keepa record book of all such drill work for future use. 


Group G 
Sierra ee / 15 'G SD deca an 
Se eOm 7 LC Ver SCs len rah hve mmCtenc islet) 


Proceed with each of the following groups as with group F, 
emphasizing the work in problems. 


| Group H 
mene. | * 5 6 Oe ee ee ae 
Sr Dia teereverses Slop rome Grey mnt 2 eriding.18°4 


eee ea 


Group I 


6 . MOTELS 2Y7 


iepere ob 4 
Zi See Teversesem a MOO i4 6 ending, is 3 
3 


Group J 


Step 1 lee Oe 7 GOL 2 
Leo Oe 4 Gtending 1s 


46 THE UNIVERSITY OF THE STATE OF NEW YORK 


Group K 
Stepetni we ou c/mO 20a). pet eee 
2.3 “4 '5 "reverses O79 ecm / Oper diame 


i ee 


Drill in addition should be kept up during the whole year for a 
few minutes each day. 
Problems: by teacher; by pupils 


Type of oral analysis acceptable from a third grade child 


Subtraction 
1 Review subtraction 
a Take up a few combinations at a time 
(1) Take up as grouped for addition include 


© oO 


(2) Give the additive form of drill, as —4 as four and ? 
make 9 

(3) Give subtraction without carrying 

(4) Give subtraction with one-step carrying, using 10 only in 
the minuend for this step as pupils have had so much 


drill on the 10 combinations, thus: 


90 90 90 990 909 9990 799990 
—88 —28 —/6 —176 —176 etc. 5753 14543 


en 


(5) Two-step carrying as 
1090 1090 1090 
—545 —953 —377 etc. 


(6) Two-step carrying and cipher difficulty 
109990 90909 
—/6042 —85443 


(7) Continuous carrying 
10900 90090 
—5643 —34584 


b Illustrations for drilling on subtraction involving the 8 ending 
group 
(1) 8 8 8 Saale 8 Sa ao 8 


(2) 


(3) 


(4) 


(5) 


(6) 


c Illustrations for drilling 


(2) 


(3) 


(4) 


ELEMENTARY 
8888 8888 
—4365 —7812 
88880 9890 
—17654 —2346 
1890 =18880 
—976 —9033 
9980 990 
—5485 —399 
10089 10908 
—7694 —6469 


group 


77/7 
—4506 


1777 
—864 


18970 
—9574 


77/7 
—2173 


WA 
—935 


178000 
—89327 


ARITHMETIC SYLLABUS 


888 


47 


—840 (no changing) 


10980 
—7045 


18990 
—9942 


9980 
— 3489 


10008 
—8949 


YUN 


7 
6 


7/17 


7 
Wh 


(changing one step) 


(changing twice) 


(consecutive carrying) 


(consecutive carrying ) 


on subtraction involving the 7 ending 


Vs 
8 


—2323 (repetition is helpful for any 


787 


troublesome pair of figures) 


8887 


—429 —1984 (changing one step) 


77/77 
—6/89 


d When the process is well established, emphasize the subtraction 
vocabulary and the meanings thereof. 


(1) Take away, minus, subtract 


(2) Less than, greater than, difference between 


(3) Minuend, subtrahend, remainder 


e Problems involving the subtraction idea in (1) and (2) above 


Multiplication 
1 A good order for taking up the tables is 5’s, 2’s, 10’s, 4’s, 3’s, 6’s, 
Se 9's 11's, e12's: 


48 THE UNIVERSITY OF THE STATE OF NEW YORK 


2 


or oaie ene oa! 


Review those taught in second grade, thus: 


Two'5’s -arewlO ho peo) Lap ae cee 


55 5 eee 

10: 5 ve Sead id 

Three 5’s are 15 HLS SESS es 
Four 5’s are 20 ZO 255 eed 
Five 5’s are 25 7 +257 med 
Six 5’s are 30 30 


Always teach each combination apart from the table so that 
children will not be forced to say the table to get required 
result. 

Teach the inversion as 3 <7, or 7 X 3. 

Teach table form but do not have it memorized in form. 

For oral drill say two 5’s are 10, three 5’s are 15 etc. 


Very early after teaching a few multiplication facts they should 
be put to use in work thus: 


52 S44 coe oae 
DEB et Pa 


ee 


Introduce cipher difficulty 
a Give plenty of drill on combination, thus 
Oo Sis Ops a eset: 


b Use cipher at end of multiplicand, thus: 


40 30 
x pe 

c Use cipher in multiplicand, thus 
402 401 


pie <5 etc. 


Multiplication with two figures in the multiplier 
Multiplication with three figures in the multiplier 
Multiplication with ciphers in the multiplier 
Teach the use of the word “times” by such applications as 
a It costs 5 cents for one bag of marbles. It costs 2 times 5 cents 
for two bags. 
It costs 5 cents for one bag of marbles. It costs 3 times 5 cents 
for three bags. 
Toys that cost four nickels cost 4 * 5 cents or 20 cents. 
Toys that cost seven nickels cost 7 & 5 cents or 35 cents. 
Toys that cost two nickels cost 2 & 5 cents or 10 cents. 


ELEMENTARY ARITHMETIC SYLLABUS 49 


b Suggested application for table of 10’s. 
Articles that cost three dimes cost 3 & 10 cents or 30 cents. 
Articles that cost six dimes cost 6 & 10 cents or 60 cents. 
Articles that cost four dimes cost 4 * 10 cents or 40 cents. 


c Suggested application for table of 3’s 
Four yards contain 4 X 3 feet or 12 feet. 
Six yards contain 6 X 3 feet or 18 feet. 


d Suggested application for table of 4’s: four quarts make a gallon 
e Suggested application for table of 6’s: cost of sugar etc. 
f Suggested application for table of 7’s: seven days in a week etc. 


g Suggested application for table of 12’s: twelve inches in a foot; 
or the dozen 


h Suggested application for table of 8’s: cost of milk ete. 


Division 


— 


Division should be taught along with multiplication, but with the 
emphasis on the multiplication and later upon the division. 


2 In teaching division make a careful grading of the difficulties and 
take up one at a time. 


a Introduce by such questions as: How many 5’s are there in 10, 
habs Soe yhal 24a 
At 5 cents each, how many pencils can be bought for 10 cents? 
b Follow drill on multiplication tables with division drills thus: 
How many 2’s in 4, in 10, in 12 etc.? 


3 Division of numbers giving no remainder 


a b C d 
ee On 4 of 6 
82 Zia. Sof 8 3 
1Q—35 5)10. tof 10 4 
fs Se 3)18 8 
4 Division of numbers with remainders 
a 52 Pyoyee 3 Ot 5 3 


b Give drill in mixed order. 

c Always bring out that when dividing by 2 the remainder, if 
there is one, must be less than 2, and when dividing by 3 the 
remainder must be less than 3, etc. 

Because a child knows his multiplication tables it does not 
follow that he knows his division tables. All children need 


. 


50 THE UNIVERSITY OF THE STATE OF NEW YORK 


on 


much practice on finding remainders. Lack of sufficient drill 
here is responsible for some of the trouble with long division. 


d Give plenty of drill on the cipher difficulty. 


Short division process 
a Use for divisor the figure representing the table being drilled 
upon. 
b At first use only even numbers 
2)486 
c Use numbers involving a remainder 
2)487 
d Numbers involving carrying 
2) 45634 
e Proof 
f Vocabulary — divisor, dividend, quotient, remainder, is con- 
tained in 
Problems 
a Application of division tables to concrete work 


b While drilling on division tables make plenty of applications to 
problems. Differentiate between the two forms of division: 
(1) Measurement 
(2) Partition 
c Type problems to illustrate measurement 
At 4 cents each how many articles can be bought for 8 cents? 
Two articles 
Analysis. There will be as many articles as there are 
4’s in 8; or 
There will be as many articles as 4 cents is contained 
times in 8 cents. 
d Applications 


(1) Table of 2’s 
How many 2-cent stamps can be bought for 5 cents? 
How many 2-cent stamps can be bought for 7 cents? 


(2) Table of 5’s 
How many nickels in 20 cents? 
How many nickels in 30 cents? 
How many nickels in 35 cents? 


ELEMENTARY ARITHMETIC SYLLABUS 51 


(3) Table of 7’s 
How many weeks in 14 days? 


7 days in one week, in 14 days there are ... weeks. 

7 days in one week, in 70 days there are ... weeks. 

7 days in one week, in 21 days there are ... weeks. 
(4) Table of 3’s 

3 feet in one yard, in 12 feet there are ... yards. 

3 feet in one yard, in 15 feet there are ... yards. 

3 feet in one yard, in 18 feet there are ... yards. 


e Partition division; part taking tables of % of, % of 
(1) Type problems 
Divide 12 cents equally between two boys; each will receive 
Y of 12 cents or 6 cents, thus 000000 000000 
(2) Applications 
(a) Table of 2’s 
If two boys divide 20 cents equally, one boy will receive 
YZ of 20 cents. 
If two boys divide 22 cents equally, one boy will receive 
Y of 22 cents. 
(b) Table of 4’s 
Divide the cost of 40 cents equally among four boys, 


one boy pays % of 40 cents. 
Divide the cost of 24 cents equally among four boys, 


one boy pays % of 24 cents. 


Third Grade Problems 
In the first and second grades an effort should be made to give 
the child 
1 A number concept 


2 A real feeling of a need for number 
In the third grade a main purpose on the part of the teacher 


should be to create for the child 
1 A real need for the combinations, processes etc. 
2 A real need for information as to prices in stores, values etc. 
3 A need for some units of measure as pint, quart, ounce, pound 
etc. 
4 A desire for skill in number with some idea as to what skill 
could be acquired by a third grade child 
The project undertaken by the class should reveal such needs to 
the child and should serve as a motive for much of the drill required. 


a THE UNIVERSITY OF THE STATE OF NEW YORK 


One project alone may not meet all the above conditions but minor 
projects undertaken by the class, situations present at the moment 
requiring the use of numbers, and problems devised by the teacher 
will all serve as motives for the drill work. 

Playing games may be used as a source of number stimulus. 


Suggestions for projects 


1° Construction projects call for measures as inch, half inch ete. 
2 Constructing a weight and height chart 
3 Constructing maps and charts call for drawing to a scale 
4 Situations in class calling for number 
a Milk supply used by school 
b Lunch bought at school 
c Class party or class picnic 
d Class scale of standings etc. 


e Store with making change, sales slips etc. 


Outline of project, “The grocery store” developed by a group of third grade 
teachers 
1 Teacher’s aims 
a A means of suggesting problems and drills for arithmetic 
b Language drill 
(1) Courtesy developed between clerk and customer 
(2) Telephone order; correct English 
(3) Abbreviations: bu., lb., pk. etc. 
(4) Vocabulary 
c Spelling 
(1) Lists to use on writing orders etc. 
d Penmanship 
(1) Written orders 
(2) Marking of goods 
e Drawing 
~ (1) Posters 
(2) Construction of material 
f Geography 
(1) Where products are obtained 
(2) Transportation 
g Physiology 
(1) Cleanliness in store 


(2) Cleanliness in homes 
(3) Beneficial foods 


ELEMENTARY ARITHMETIC SYLLABUS 53 


h Reading 
(1) Problems 
2 Children’s aims 
a Efficient clerking and shopping 
b Fun playing 
3 Materials 
a Tables or shelves 
b Money, toy or real 
c Measures, weighing scales, pint and quart measures, gallon, 
bushel, yard, foot etc. 
d Supplies or stock . 


4 The store as a device for drill gives motive for 
a Practice in the four fundamental operations 
b Original problems by class 
c Drill in making change 
d Drill in measures : 
e Drill in making sales slips, bills etc. 


5 Types of problems used in the store project 

a If Mary bought three boxes of matches at 6 cents each, how 
much would they cost? 

b Francis went to the store with 50 cents. He bought a peck of 
apples at 22 cents. How much change did he receive? 

c If Helen went to the store and bought a loaf of bread at 11 
cents, a pint of milk at 6 cents and a pound of butter at 
50 cents, how much did she pay? 

d How much change from a dollar would I receive if I bought 
groceries amounting to 57 cents? 

e If a quart of vinegar costs 12 cents, how much will a pint cost? 


Problems 
Problems arranged by a group of third grade teachers illustrative 
of types of reasoning in the four fundamental processes. 


Addition 

1 From the menu card John chose for lunch lamb stew at 12 
cents, spinach at 6 cents, mashed potatoes at 4 cents, bread and 
butter at 2 cents, prunes at 3 cents. Mary, his sister, chose the same 
but added ice cream at 5 cents. John paid for both. How much 
did it cost him? 

2 The oatmeal for a breakfast cost 8 cents, the milk 4 cents, the 
fruit 10 cents, the rolls and butter 5 cents, and the eggs 8 cents, 
How much did this food cost? 


54 THE UNIVERSITY OF THE STATE OF NEW YORK 


3 The meals for a small family cost $1.70 on one day and $2.20 
on another day. How much did they cost for these two days? 


Subtraction 


1 If you buy something for 7 cents at the store and give the clerk 
10 cents, what change should you receive? 


2 John has 8 cents. How much more does he need to buy a 
—15-cent ball? 


3 Charles had 12 marbles, and lost 7 of them. How many had 
he left? 


4 Fred is 9 years old, and his cousin is 15 years old. Fred is 
how much younger than his cousin? 

5 Walter had 63 marbles. After losing 18 how many had he 
remaining ? 

6 James weighs 86 pounds and Henry weighs but 68 pounds. 
Find the difference in weight. 


7 A boy sold 20 papers on Monday and 15 on Tuesday. How 
many more did he sell on Monday than on Tuesday? 


Multiplication 
1 What is the cost of 6 rubber balls at 8 cents a ball? 
2 If it is 3 weeks until vacation how many days is it? 


3 If a boy earns 9 cents each morning delivering papers, how 
much will he earn in 3 mornings? 


4 At 8 cents each, how much must your mother pay for 6 grape 

fruit? 
Division 

1 At 2 cents each, how many pencils can you buy for a dime? 

2 Robert has 50 cents. How many tablets can he buy at 10 cents 
apiece ? 

3 If you divide 14 pieces of candy equally among 7 children, how 
many pieces will each receive? 


4 Fifteen pennies are to be divided equally among 3 boys. What 
part is each to get? How many pennies will each have? 


5 Mary had 12 candies. She gave May % of them. How many 
had May? 


6 If I have 12 books, what is % of them? 


ELEMENTARY ARITHMETIC SYLLABUS 55 


Measures 
Liquids 
1 From 2 gallons of milk how many quarts can be sold? How 
many pints? 
2 Mary’s mother put up 16 pints of grape juice. How many 
quarts was that? How many gallons? 


Dry 
1 Henry gathered 16 quarts of nuts. How many pecks did he 
gather? 


2 If I feed my horse a peck of oats a day, how long will 1 bushel 
last? How long will 6 bushels last? 


Dozen 
1 What will half a dozen eggs cost at 2 cents each? 


2 What will one dozen of oranges cost at 5 cents apiece? 


Avoirdupois 
1 Robert bought % pound of tea. How many ounces did he buy? 


2 Ruth has 2 boxes of candy. Each weighs 8 ounces. How many 
pounds of candy has she? 


Linear 
1 A schoolroom is 30 feet long. How many yards long is it? 
2 How many feet are there in 6 yards? In 9 yards? 


Time 
1 There are 7 days in 1 week. How many days in 3 weeks? 
How many school days in 3 weeks? 
2 A man pays $5 a month for the use of a garage. How much 
does he pay a year? 
3 Frank was to play with James an hour. When was his time up, 
if he reached James’ house at 4 o’clock? 


United States Money 
1 Find the cost of 5 pencils at 1 cent each. 
2 What change will be left if I buy 25 cents worth of candy and 
give the clerk 50 cents? 
3 Ruth bought an apron for $1 and 2 handkerchiefs for 25 cents. 
What was the total cost? 


56 THE UNIVERSITY OF THE STATE OF NEW YORK 


FOURTH GRADE 
Condensed Outline 
The teacher should read carefully the general introduction to the 
syllabus and should be familiar with the contents of the syllabus for 
the third and fifth years. ; 


First Half 
A Integers 


I Arabic notation and numeration through hundred thousands 
II Roman numerals to L 
III Addition 
IV Subtraction 
V Multiplication tables 
VI Multiplication process 
VII Division tables 
VIII Short division 


B Fractions 


I Part taking tables 
II Simple fractions 


C Measures. Dozen, ounce, pound, pint, quart, gallon, foot, inch, 
yard, United States money, peck, bushel 


D Problems 
I Applying each of the above topics 


II One-step problems 
III (Optional) Long division 


Second Half 
A Integers 
I Arabic notation and numeration through millions 

II Roman numbers to C 

III Addition 

IV Subtraction 

V Multiplication 

VI Long division with two and three-digit divisors 
B Fractions. Very simple fractions 
C Measures. Measures and United States money 
D Problems 


I One-step problems completed 

II Some two-step problems 
III (Optional) Long division with three-digit divisors 
IV Multiplication and division of fractions 


ELEMENTARY ARITHMETIC SYLLABUS 57 : 


Expanded Outline 
A Integers 
I Notation and numeration through hundred thousands. Apply 
to geography for reading population, heights etc. 
If Roman numerals to L. Apply to reading chapter numbers 
of books. 
III United States money 
IV Four fundamentals 
1 Addition 


a Study syllabus for second and third grade addition and 
use for review. 
or 
b Select some other scheme of grouping combinations and 
review. 

(1) Take a few combinations at a time and concentrate 
on them, using them in series drill and in column 
addition. 

(2) Check all work by adding columns in reverse order. 

(3) Terms. Add to the child’s vocabulary such terms 
as sum, addend, addition, amount, column. 

(4) Tests. Keep work up to standard test by Courtis’, 
Woody or other standard tests. 

(5) Problems 

(a) From some good text 
(b) Original problems given by pupils 
(c) Some project undertaken by the class 


2 Subtraction 

a Review. Study syllabus for second and third grades. 
Give special attention to the following difficulties: 
changing in subtrahend and minuend, ciphers in minu- 
end, and any troublesome combination. 

b Reteach proof for checking work. 

c Terms. Add to the child’s vocabulary so that he shall 
use with facility the terms subtraction, minuend, sub- 
trahend, remainder, difference between, minus, less 
than, greater than. 

d Drills involving subtraction at sight are a good prepara- 
tion for long division. 

e Tests. Keep grade up to standard. Courtis’ tests or 
other standard tests should be used at least twice a year. 


58 


THE UNIVERSITY OF THE STATE OF NEW YORK 


f Problems. All types of difficulties; see third grade 
syllabus. 
Acquiring skill in making change should be part of 
this year’s work. 
g Sales slips. Bring in sales slips from local business con- 
cerns. 
Teach child to make out a sales slip and to total. 


Multiplication 
a Review all tables 

(1) Time tests. Speed for complete table should be 12 

seconds or less when answers only are given. 
(2) While drilling on tables teach terms, multiples and 
factors. 

(3) Apply problems to tables; see third grade syllabus. 
b Multiplication 

(1) By 2 and 3 figures in multiplier 

(2) By multipliers with ciphers 

(3) By multipliers with ciphers at end 

(4) Ciphers in multiplicand 
c Proof. At this time by repeating work, later by division. 
d Terms. Times, multiplication, multiplier, multiplicand, 

product. . 
e Time tests. Use Courtis’ test or other standard tests. 
Division 
a Division tables, no remainders thus 
PA WA cio bea 5 YZ of 12 172 
Emphasize which is the dividend in each case. 


b Division tables with remainders 
c Short division with carrying 
d Problems applying. Measuring, see syllabus page 55 
third grade; partition, see syllabus page 54 third grade. 
It is suggested 20 per cent of recitation period be 
devoted to review of fundamental and oral problems 
already completed. 
e Long division 
(1) Take one difficulty at a time. 
(a) Begin with very easy examples so that only the 
process will require attention. 
(b) Use as divisors 21, 31, 41, 51, 61 etc., then 22, 
32, 42 etc. omitting all divisors from 13 to 19 


ELEMENTARY ARITHMETIC SYLLABUS 59 


for the present. They are more troublesome. 
Take care to have divisor contained in divi- 
dend the trial number of times. 


(c) Divisor contained in first two of the dividend 
thus 
ol 
21)651 
63 


21 

21 
Plenty of oral drill should be given at this 
point for deciding quotient figure by giving 
examples with one-digit quotient for drill as 


21) 64 2b) (2 w etc. 


(d) Division with remainder thus 


Z, 
31 21 
21)653 
63 


23 
zt 


Z 


(e) Divisor contained in dividend of four figures 
thus 


311 
21)6531 
63 


v6 
21 


21 
21 


60 THE UNIVERSITY OF THE STATE OF NEW YORK 


(f) Divisor contained in first three figures of divi- 


dend 
657 iy S331 
ZV os on 31) 1046 
126 93 
119 or 116 
105 93 
147 5 
147 


(g) Ciphers in quotient 


108 
21)2268 
21 


168 
168 


(h) Divisor contained in first partial dividend less 
than the trial quotient figure 
6 (not 7) 
41)284_ 
246 


—E 


(2 ae LOL 


(3) Terms. Is contained in, (not goes in) division, 
divisor, quotient, remainder 


(4) Problems 


(a) Aim at this time to have children able to recog- 
nize the process involved in any one-step 
problem, whether it be addition, subtraction, 
multiplication, or division. 

(b) When the problem is solved allow the child to 
state the solution in his own words rather 
than in a set form. 

(c) Often after stating the solution have the child 
say: “This is a problem in addition,” or 
“ subtraction,” etc. 

(d) Often have pupils recall all problem situations 
that require the process being studied, thus: 
As finding gain requires subtraction, etc. 


ELEMENTARY ARITHMETIC SYLLABUS 61 


(e) Have pupils make up problems in the process 
being studied. 

(f) Formulate problems without numbers as: If 
you know the cost of the groceries and the 
meat how will you find the total cost? 

(g) Aim to teach the problem vocabulary as total, 
average per week, profit, etc. 

(5) Application to averages, sharing costs ete. 

(a) Average age 

(b) Average weight 

(c) Average standings for girls or boys in spelling 
and arithmetic 

(d) Average temperature outside for a weal a 
month 

(e) Average attendance for a week 
Costs for party, sleigh ride, picnic 


B Fractions 


It is suggested the teacher read in Psychology of Arithmetic by 
Doctor Thorndike, Knowledge of the Meaning of a Fraction. 

The following introduction by Doctor Thorndike shows the various 
steps a child must take in grasping a full understanding of fractions. 

Scientific teaching now builds up the total ability as a fusion or 
organization of lesser abilities. What these are will be seen best by 


examining the means taken to get them. 


1 First comes the association of 4 of a pie, 5 of a cake, 4 of an 


apple and such like with their patie meanings ee cating: The 


is taught 1 pie = two —’s, three —’s, four —’s, five —’s, six —’s, 
. 5 4 5 6 
etc. similarly for 1 cake, 1 apple and the like. So far he under- 


stands fractions as simple parts of the units. 

2 Next comes the association with 4 of an inch, 4 of a foot, 4 of 
a glassful where the derivation of the fraction from the unit is not 
so obvious. | 


3 Next comes the association with 4 of a collection of eight pieces 
of Gy, 3 of a dozen of eggs, 4+ of a squad of ten soldiers etc. 
until $, 4, 4 etc. are understood as names of certain parts of a collec- 


tion of objects. 
4 Next comes the similar association when the nature of the 


collection is left undefined, the pupil responding to 4 of 6 is ... 
BeTIOtIG apt eS HOLM pare UST OL, <2, 


a! 


62 THE UNIVERSITY OF THE STATE OF NEW YORK 


Each of these abilities is justified in teaching by its intrinsic merits 
irrespective of its later service in helping to constitute the general 
understanding of the meaning of a fraction. The habits thus formed 
in grades 3 or 4 are of constant service then and thereafter, in and 
out of school. 

5 With these comes the use of 1/5 of 10, 15, 20 etc., 1/6 of 12, 
18, 42 etc., as a useful variety of drill on the division tables valuable 
in itself, and a means of making the notion of a unit fraction more 
general by adding 1/7 and 1/9 to the scheme. 

6 Next comes the connection of 3/4, 2/5; 3/5; 4/5, ‘2/3, 1/6, 
5/6, 3/8, 5/8, 7/8, 3/10, 7/10, 9/10 each with its meaning as a 
certain part of some conveniently divisible unit; and 

7 Connections between these fractions and their meanings as parts 
of certain magnitudes; and | 

8 Collections of convenient size; and 

9 Connections between these fractions and their meanings when 
the nature of the magnitude or collection is unstated as in 
Ay Ofsl Die een 

It is important that the teacher recognize the different conceptions 
of the fraction, for the child who has been taught concretely the 
fraction idea expressed in 2 may fail seriously when the same frac- 
tion idea is met in connection with the collection as in 3 etc. 


I Informal fractions 
Before the fourth year some knowledge of fractions has 
already been acquired, as the part taking tables with its appli- 
cation to division or some written symbols are familiar. 


II Formal fractions; outline of procedure 


1 Create a need for a knowledge of fractions. The following 
topics are suggested as sources of problems or projects 
involving fractions which will challenge the pupil to the 
new knowledge. 

a Drawing to a scale as plotting the garden, drawing» maps to 
a scale 

b The store, dry goods or meat market will require operations 
with fractions. Sales tags may be made out for rem- 
nants etc. 

c Recipes for cooking may be halved or doubled. This will 
involve fractional parts. 

d Construction. Making toys or other articles involving use 
of ruler. 


ELEMENTARY ARITHMETIC SYLLABUS 63 


2 Limit the fraction field to fractions with small denominators 
and drill thoroughly on the fractions common in business 
usage. See table of fractions at end of fourth year outline. 


3 Order of work 
Study fractions one at a time in the following order: 
halves, 3ds, 4ths, 5ths, 6ths, 7ths, 8ths, 9ths, 10ths etc. 
Beginning with halves take the fraction through the follow- 
ing experiences before attempting 3ds, and after com- 
pleting 3ds take up 4ths. 
a The unit, as 1 yard of ribbon 
b Fraction 1/2 yard, 2/2 yards etc. 
c Combination of unit and fraction or the mixed number as 
11/2 yards 
Reduction of integer to fraction and vice versa 
ime 1 ep Ta kee taf 
avAddition 1/2-1/2—=2/2Z= 1 or 
1 half 1/2 
1 half ee 


2vhalves 4-2/2 
e Subtraction 4/2—2/2=2/2=1 
Subtraction 2/2—1/2=1/2 

f Part taking table 
g (Optional for fourth grade) Multiplication of a fraction by 

aimintereri2 3 L/ 2 
h Division, like denominators only, as 

Py ONY i tomes 7A 

i Fractional equivalents 

Delay these until after a study of fourths or even later 
until several equivalent fractions are known. 

Reduction to higher and lower terms with principle 
multiplying or dividing both terms by the same factor 
does not change the value of the fraction. 

j Addition of unlike fractions 
k Cancellation applying paragraph 7 
! Fractional parts 
(1) Part taking table 
(2) Relation of part to whole (2 is what part of 4?) 
m Concrete problems at each step 
n Original problems at each step 


64 THE UNIVERSITY OF THE STATE OF NEW YORK 


Illustrations for fourths 

1 The pie or cake where the fraction is clearly a simple distinctive 
Part OLeadeUlite weit ig suggested that the class do the cutting and 
folding and save in envelopes for future work. 

2 The dozen (illustration for fractional part of a collection. Use 
actual collections of materials and drawings.) 

000 000 000 000 

_3 The rectangle. Good for cutting and folding lesson to illus- 
trate a fraction as a distinct part of a unit as 1/4 of a chocolate bar 
or to illustrate a fraction as a part still involved in the unit as 1/4 of 
a glassful. 


A 4/4 B 3/4 Oro yaee D 1/4 


4 Foot or yard as a unit of measure, where the fraction is within 
the unit and is not so clearly differentiated as the separate piece of 


pie. But a very common use of the fraction. 
A 4/4 


| 


Suggestions follow for developing each topic using the above 
illustration for fourths. 

Use letter names 

Use fraction names 


1 Idea of unit, one yard 
Find the unit. Name it. One yard 
Draw other units equal to this one 
What is a unit?) Name other units. 


2 Fraction idea . 
Into how many parts is the yard divided? Name the parts. 
Write the name of parts. Find Bon A. B is what part of A? 
Write the fraction name. Find C on A. Cis what part of A? 
Write the fraction name. 


ELEMENTARY ARITHMETIC SYLLABUS 65 


3 Combination of whole number and fraction 
Reduction of integer to fraction 2 = 4/2 
Reduction of mixed number to fraction 21/2 = 5/2 
Draw a unit equal to A or draw a yard and divide into fourths. 
How many fourths in two yards? 2 = 8/4 
Write expression. 
How many fourths in 21/4 yards? in 22/4 yards? in 2 3/4 
yards? 
How many fourths in 3 yards? in 3 units? in 31/4 yards? 
in 32/4 yards etc. 
Give plenty of drill of this kind with written expression on board. 
Also reverse question and ask: 4/4 yards make how many yards? 
how many units? 4 are how many yards? 
Change 12/4, 17/4, 5/4, 7/4, 6/4, 8/4 ete. 
Give plenty of drill on this point using illustrations until the idea 
is clear. Then give drill without illustrations from board as 


follows: 
4 = 5 = 16/4 = s/t 
41/4= ey Bie Vi LA 
42/4= 52/4 = 12/4 = 10/4 = 
5374: 4/4 = 213/4 = 
4 Addition 


Add lines B and C, page 64, using fraction names 

3/4 + 2/4=5/4=11/4. 
Add lines B and C, page 64, using word names 
Write on board expression for the above number symbols. 
Add lines C and D using fraction names and word symbols. 
Express on board. 


2/4 + 1/4 = 3/4 also 2/4 also 2 fourths 
1/4 1 fourth 


3/4 3 fourths 
Add all the fourths in the illustration. 
And express on the board. 
4/4 
3/4 
2/4 
1/4 


10/4 =22/4 


Give many problems under (4) using several units in the 
illustration. 


66 


5 


6 


THE UNIVERSITY OF THE STATE OF NEW YORK 


Subtraction 
If 2/4 are taken from 3/4 what is left? 


Write expression for this as 3/4 also 3 fourths 
—2/4 —2 fourths 


1/4 1 fourth 

Take 1/4 from 3/4 

Take 1/4 from 4/4 

Many problems like the above using 2 units, 3 units, etc. 
(Optional in fourth grade) Multiplication of a fraction by an 

integer 

a Find a line 2 times D. 

Find a line 3 times D. 

Find a line 4 times D. 

Find a line 2 times C. 

See page 64. 


b Find a line 2 times D giving fraction name. 2/4 
Find a line 3 times D giving fraction name. 3/4 
Find a line 4 times D giving fraction name. 4/4 
Find a line 2 times C giving fraction name. 4/4 
c Find a line 2 times 1/4. Write the full expression on board 
2X 1/4=2/4 2a etOuLto = routs: 
Find a line 3 times 1/4. Write and name it. 
Write expression'on board) 3>< 41/4’ == 3/4" 3)>o) fourth 
3 fourths. F 
Give many similar problems under each group. 
(Optional in fourth grade) Division, using only like fractions 
a How many times can you find D on B? 
How many times can you find D on A? 
How many times can you find C on A? 
b How many times can you find 1/4 in 2/4? 
How many times can you find 1/4 in 3/4? 
How many times can you find 1/4 in 4/4? 
c Repeat group 0 writing the expression. 
How many times can you find 1/4 in 2/4? 2/4—1/4=2 
How many times is 1/4 contained in 3/4? 3/4~+1/4=3 
How many times is 2/4 contained in 4/4? Write expression 
4/4—-2/4=2 
Fractional equivalents ; 
a Take up the 1/2 series with illustrations. 
Take up the 1/3 series with illustrations. 


ELEMENTARY ARITHMETIC SYLLABUS 67 


Take up the 1/5 series with illustrations. 

Take up the 1/6 series with illustrations. 

Take up the 1/7 series with illustrations. 

Carry this as far as necessary to make clear the underlying 
principle. 

b Establish by illustration the series of names for 1/2 

Dee? [4 =a /S = Of 1616432. 

c Establish the same way the series 

Pa 2/03 3/9 ele 1S. 

d Take the fraction and multiply both numerator and denom- 

inator by 2, by 3, by 4. 

What is the answer in each case? 

How do these answers compare in value? 

What is the effect on the value of the fraction of multiplying 
both terms by the same number? 

Establish that multiplying both terms of the fraction by the 
same number does not change the value of the fraction. 

e In the same manner, taking the fractions expressed in higher 
terms show that dividing both terms by the same number 
does not change the value of the fraction. 

f After much drill from illustrations and without illustrations 
in deduction of fractions, teach 


9 Addition of fractions having unlike denominators 


10 Cancellation with application of the principle that dividing both 
terms by same factor does not change value of fraction. 


11 Fraction parts 
a Part taking 1/4 of 2/4 of etc. 
(1) Part taking 
This is application of table already taught. Let A=8 
then find 1/4 of A, 2/4 of A, 3/4 of A is 6. 
b (Optional in fourth grade) Relation of part to part; 2/4 is 
what part of 3/4? 
Relation of part to whole; 2 is what part of 4? 
(1) Relation of part to part 
(a) Write letter names, point out 
D is what part of C? Answer: D is 1/2 of C. 
D is what part of BP? Answer: D is 1/3 of B. 
C is what part of B? 
D is what part of A? 
C is what part of A? 


68 THE UNIVERSITY OF THE STATE OF NEW YORK 


B is what part of A? 
See page 64. 
(b) Find and point out 
(1) 1/4 of 2/4 
one fourth equals what part of two fourths? 
(2) 1/4 of 3/4 
one fourth equals what part of three fourths? 
(3) 2/4 of 4/4 
two fourths equals what part of four fourths? 
(4) 2/4 of 3/4 
two fourths equals what part of three fourths? 
(2) Relation of part to the whole using integers, as, 3 is what 
part of 4° etc. 


12 Problems 


a Use social and economic environment of child for problems. 
Use problems relating to meat market, grocery store, depart- 
ment store, making Christmas gifts, scores etc. 

b Original problems by class 

c ‘Textbook problems 

d Projects worked out by class 

e Athletic records 


13 Time allotment 


1/3 oral work (drill work) including fundamentals 

1/3 discussion of problems 

1/3 written 

Teach children to check up their work for errors, by proofs; 
by looking over work the second time; by testing approxi- 
mate answers to see if results are reasonable. 


14 Tests 


a To standardize the class work Courtis’ fraction tests may be 
used, in case no standards are obtainable. 

b In all drill in the various processes set up standards of speed 
and accuracy. | 

c Children should know what these standards are and teacher 
should endeavor to arouse enthusiasm of children to attain 
or surpass the standards. 


A Grocery Store Project 
(Arranged by Miss E. Scheyer, Dunkirk, N. Y.) 


Work up enthusiasm over a grocery store. Make children realize 
that it is necessary for someone to go to the store daily. Talk about 


ELEMENTARY ARITHMETIC SYLLABUS 69 


what they give in exchange for groceries. 

1 Get class to suggest project. 

2 Motive in general: To teach children the industrial life by which 
they are surrounded and of which they are an active part. 


3 Teacher’s aims 


a To help the children find vital problems 

b To teach them an appreciation of industry 
c To develop social instinct 

d To develop leadership 

e To develop cooperation 

f To give information 

g Yo develop thought and reasoning 

h ‘To increase the power of concentration 


4 Children’s aims 


a To learn to make purchases 

b To learn United States money 

c To add, subtract, multiply and divide quickly and accurately 
d To learn fractional parts 

e To learn measures 


5 Subjects 


a Language — oral and written 
b Arithmetic 
c Geography 
d Spelling 
e Penmanship 
f Drawing 
g Why can grocers accept a smaller margin of profit on a pound 
of sugar than on a pound of tea? 
Elementary meaning of overhead costs. 


6 Outline by subjects 

a Language — oral 
(1) Class discussion of plans for store 
(2) Experiences in buying | 
(3) Value in paying cash as opposed to charging 
(4) Value in paying cash and carrying 
(5) Discussion of personal visit to store 
(6) Increase vocabulary 

b Language — written 
(1) Written record of purchases 


70 THE UNIVERSITY OF THE STATE OF NEW YORK 


(2) Written advertisements for daily paper and windows. 
Secure names of groceries nationally and locally known. 

(3) Employment of boy in grocery — hour’s pay, what to know. 

(4) Food rationing and substitutes 

c Arithmetic 

(1) Study and use of United States money 

(2) Addition, subtraction, multiplication, division, and fractions 

(3) Sales slips, bills 

(4) Problems 

(5) Measures 

(6) Losses by poor accounts 

(7) Losses by perishable products 


Type examples 


1 Ruth paid 58 cents for butter and 50 cents for tea. 
She gave the clerk $1.25. How much change did she 
receive ? 
2 Mary has 25 cents. Eggs are 32 cents. How much 
more money does Mary need? 
3 There were 100 bananas in a bunch. After 48 of them 
had been sold, how many bananas were left? 
4 Find the cost of 2 dozen eggs at 32 cents a dozen. 
3 Find the cost of 2%4 dozen eggs at 32 cents a dozen. 
6 John has 40 cents. How many 5-cent measures of 
chestnuts can he buy? How many &-cent measures? 
7 There are 4 quarts in a gallon. How many gallons in 
80 quarts? 
8 Mrs Smith bought 144 clothes pins, how many dozens 
did she buy? 
9 If a gallon of maple sirup costs $1.12, how much should 
a quart cost? 
10 A grocery sold 25 barrels of flour for $200. Find the 
price per barrel. 
11 At 32 cents a dozen, what does the grocer receive for 
3 eggs, 9 eggs? 
12 When you buy 3 lemons for 10 cents what is the cost 
for one? 
13 At 45 cents a pound what will 9 ounces of cheese cost? 
d Geography 
(1) Talk about oranges, spices, cocoa, cocoanut, and other 
things sold. 
(2) Tell where they come from. 


ELEMENTARY ARITHMETIC SYLLABUS 71 


(3) Tell how they are obtained, bringing in a study of transpor- 
tation. 
(4) Tell what we give in return bringing in a study of exports, 
imports. 
(5) Talk about countries from which coffee, tea, spices, cocoa 
come. 
Study people in these countries and how they differ from 
our own people. 
e Spelling 
Difficult words used in subjects such as transportation, coffee, 
cocoa, oranges, export, import. 
f In connection with language 
Writing of sales slips 
g Health and hygiene 
(1) Clean grocery store 
(2) Supervision by health authorities 
(3) Supervision of weights and measures 
h Drawing 
(1) Cut pictures of groceries from magazines and mount on 
oak tag. 
(2) Make toy money from oak tag. 
(3) Make designs for advertisements. 
1 Table of measurement 
(1) United States money 
(2) Liquid measure 
(3) Dry measure 
(4) Avoirdupois weight — ounce, pound 


Milk Situation 
Milk served in the school and sold to the children at 3 cents a 
half pint was made an interesting source of school work for a 
fourth grade by Miss B. Thompson, Dunkirk. A few of the sugges- 
tions are given here. 
1 Geography 
a Source of milk, the dairy, local farms with cows, buildings, feed, 
shelter. New York State regions noted for dairying. United 
States regions noted for dairying. 


2 Arithmetic — for discussion only 
a Some idea of capital invested, value of farm, stock, buildings, 
and implements 


72 THE UNIVERSITY OF THE STATE OF NEW YORK 


b Overhead expense 
Farm help, bottles, delivery auto, etc. 
3 For study 
a Pints, quarts, gallons 
b Fractions 1/2, 1/4, 1/8, 1/16 
c United States money 
d Days, months, weeks 
e Addition, subtraction, multiplication, division of 
(1) United States money 
(2) Fractions 
f Keeping simple accounts 


Examples of Problems Involved 

1 At 3 cents a half pint, find cost of 1 pint of milk, of 1 quart. 

2 Find cost of milk for one school week for one child, for entire 
grade. 

3 Find cost of milk for 1 month of 20 days. 

4 At 3 cents a half pint, how many half pints can be bought for 
15 cents, 30 cents? 

5 A milkman pays 32 cents a gallon for milk and sells it at 12 
cents a quart, how much does he make on one gallon? 


FIFTH GRADE 


The teacher should read carefully the general introduction to the 
syllabus and should be familiar with the contents of the sya bus 
for the fourth and sixth years. 


Condensed Outline 
First Half 
A General review of integers 
B Common fractions 
I Reduction 

II Addition 

III Subtraction 

TV Multiplication 

V Cancellation 


Second Half 
A Common fractions continued 
I Division of fractions 
II Fractional relationships 
III Decimals 


ELEMENTARY ARITHMETIC SYLLABUS 73 


B Decimals 
I Numeration and notation 
II Reduction 
III Operations 
IV Aliquot parts 


C Denominate numbers (review of tables learned) 
D Business forms 
E Problems 


Expanded Outline 


Fifth Grade (First Half) 
A Integers 

A marked increase in speed and accuracy should charac- 
terize the fifth grade work in fundamentals. 

Use some standard tests and compare class with these 
standards. 

Follow up with much drill on whatever is needed. Some 
systematic drill such as is outlined in the lower grades is 
much better than scattered drill but such systems of drill 
as the Courtis Practice Tests published by the World Book 
Company; the Studebaker Drills published by Scott Fores- 
man and Company; Compact Efficiency Drills by Iroquois 
Publishing Company are time savers, and should be used if 
available. 


| 


Notation and numeration 
II Addition 
III Subtraction 
IV Multiplication 
V Short division 
VI Long division with 2 and 3 digits in divisor 
VII Review multiplication tables, factors, and multiples 
VIII Review division tables with remainders 
IX Review part taking tables 
X Problems. Two-step problems applying to the above (see 
Problem studies, pages 8-10) 


B Common fractions 


See fraction work of fourth grade. It is suggested that 
much objective work be given followed by abstract drill 
using small denominators. It is also suggested that some 
class activity or situation involving fractions be introduced 


74 


om 


II 


III 
iY 

V 
V 


— 


VII 


THE UNIVERSITY OF THE STATE OF NEW YORK 


at this time. As each difficulty is met and solved give much 
drill wherever needed. See activities at end of outline for 
fifth grade. 

Set standards of accuracy and speed for each process in 
fractions. 


Teach fraction idea in contrast to unit. Have both the 
unit and fraction before the child and draw comparison. 
Reduction 


1 Intevers*to’ fractions 741 == 220242 i ci 
2 Mixed numbers to fractions: 11/2 — 3/2, 21/2 = 5/2, 
CUC. 
3 Fractions to whole or mixed numbers: 4/2 = ? 6/2=? 
FILE ete 
4 Fractions to higher or lower terms: 1/2 =2/4= 4/8 = 
8/16 etc. 
5 Principles involved 
a Multiplying the numerator by an integer multiplies the 
fraction 
b Multiplying the denominator by an integer divides the 
fraction 
c Dividing the numerator by an integer divides the 
fraction 
d Dividing the denominator by an integer multiplies the 
fraction 
e Multiplying or dividing both numerator and denom- 
inator by the same number does not change the value 
of the fraction 
f Only like fractions can be added or subtracted 


Addition 

Subtraction 

Cancellation 

Divisibility by 2, 5, 10; factors and multiples as needed 
for the denominators in use 


Multiplication 

1 Fraction by an integer 

2 Fraction by an integer using cancellation 
3 Mixed number by an integer 

4 Fraction by a fraction 

5 Fraction by a fraction with cancellation 
6 Mixed number by a mixed number 


ELEMENTARY ARITHMETIC SYLLABUS 75 


Sixth Grade (Second Half) 
Common Fractions Continued 
VIII Division 


It is suggested that no explanation of the process of 
division of fractions be attempted. Simply show that the 
result is obtained by inverting the divisor and multiplying. 
Teach as a mechanical process. 

IX Relation of numbers. Use small numbers in the two fol- 
lowing easy types: 
1 One number is how many times another ? 
Problem application: if 3 lemons cost 10 cents what will 
a dozen cost? 
2 One number is what part of another? 
Problem application: 2 examples missed out of 5 means 

2/5 of the lesson missed. 

X It is recommended that in all work with fractions only 
the common business fractions be used. 
eel Se Waly ed Ae gues) ead heey ee) Onno Oi. 
OO On OFA) Orel PLL ems Ue eh ee 

C Decimal fractions 
I Review of United States money 
II Numeration and notation of decimals 
1 Limit of 3 decimal places 

a Practice in reading decimals 
b Practice in writing decimals 

(1) Begin by writing as fractions 3/10 etc. 

(2) Write as equivalent decimals 3/10 = .3 etc. 

(3) Learn 1 place means tenths 

2 places means hundredths 
3 places means thousandths 
(4) Learn converse 
tenths means 1 place 
hundredths means 2 places etc. 
III Addition of decimals 
IV Subtraction of decimals 
V Reduction of decimals to common fractions. Write as 
decimals and rewrite as in form of regular fractions 
and reduce to simplest form. 
1 Pure decimal as .25 
2 Mixed decimal as 4.5 


76 THE UNIVERSITY OF THE STATE OF NEW YORK 


3 Decimals ending in fractions .02%. This form should 
be introduced just before taking up aliquot parts. 


VI Multiplication 

1 In multiplying decimals develop the rule for pointing off 
by writing the decimals as common fractions and 
multiplying as such. Observe the denominators of the 
product. Finally teach that there will be as many 
places in the product as the sum of the places in the 
multiplicand and the multiplier. 

2 Show that multiplying by 10 moves the decimal point 
one place to the right; 100 two places etc. Give plenty 
of drill on this. 


VII Division of decimals in 3 steps 


It is suggested that the pointing off in division of 
decimals be taught mechanically and the explanation as 
here developed be left until the sixth grade. 


Step 1. Decimal by an integer 


Teach pupils to place decimal point in quotient directly 
above the decimal point in the dividend before beginning 
to divide, then divide as in whole numbers. 


2) 05 sD fe): 
Step 2. Integer by a decimal 
Rewrite the decimal so the divisor is a whole number. 
Then place decimal point in quotient as in step 1. For 
example: 


27a rewritten 02.) Ase 


Step 3. Decimal by a decimal 
1 Rewrite the decimals 
a Making the divisor a whole number 


b Moving the decimal point in the dividend to the right 
the same number of places as was contained in 
the divisor 

c Placing the decimal point in the quotient over the 
decimal point in the dividend and dividing as in 
whole numbers. Insist on placing the decimal 
point before dividing. It is suggested that instead 
of checking the decimal point or crossing out the 


ELEMENTARY ARITHMETIC SYLLABUS 77 


decimal point to facilitate division of decimals, the 
example be rewritten when making the divisor a 
whole number. 


.04).156 rewritten 4)15.6 
VIII Problems involving fractions and decimals 
IX Short processes in division 
Purchasing by 100 or by 1000 
X Aliquot parts of $1 with Cecimal equivalents 
1 The pupil should know how to find the aliquot part. 
2 Learn the following and their equivalents: 
1/2 LS rl fe oat / 8; SLO Re 58574505 /8;"5/8; 
TT AWA 
Other aliquot parts may be used but it is advised 
that only those of real value in class work be taught. 
3 Application to problems 
a Sales when marked 1/3 off etc. 
b At 12 1/2 cents each, how many articles can be bought 
for a dollar? 
D Denominate numbers 
Collect denominate numbers previously learned into tables thus: 


De mea ELC: Lot Zac emis 
Sgt ec 2000 Ibs. = 1 ton (new) 
Depts aol ct. 100 lbs. = 1 cwt. (new) 
SSCS eee Pal, 
S qtsa=- 1 pk. 
Anokse:—al (bu, 
5¢ = 1 nickel Ja ae LAK 
10¢=. Ie dime 2A hr — lisday 
50¢ = 1 half dollar 60 oming=* hr, 
25¢ = 1 quarter dollar OD secanasnlermin, 
100¢ = 1 dollar DAPI teen Yt: 
10 dimes = 1 dollar 
12s ee OLOSS Optional 
1 1 doz. Surface measure 


E Business forms 
I Sales slips 
II Bills, integers or part-taking fractional used 
III Simple account keeping showing debit and credit side 


F Arithmetical information pertinent to fifth grade | 


78 THE UNIVERSITY OF THE STATE OF NEW YORK 


I Knowledge of denominate numbers 
II Knowledge of various household costs 
1 Common commodity values 
2 Average cost of food, clothing, shelter for a child, for a 
week, a month or a year 
3 Cost of school supplies for a child, for each grade, etc. 
III Budget idea, personal accounts, bills 
IV Knowledge in a simple way of what systematic saving 
amounts to 
1 Through keeping a simple bank account 
2 Through keeping a school bank account 
3 Figuring interest so many cents on the dollar 
V Knowledge of post office rates, freight rates, etc. 
G Problems from text applying above. For suggestions as to. prob- 
lems see syllabus pages 8-10. 
I Review one-step problems to keep up recognition of process 
needed 
II Two-step problems 


III Teach children to think problems through in the steps of 
the solution, thus: 


1 Given 
2 To find 
3 Solution; process 


A rigid insistence upon writing out every problem in 
the given steps is unnecessary. The easy problems will 
take care of themselves. For the difficult or more 
involved problems, the above steps will give the child 
a method of thinking the problem through. 

IV Problems from school activities 


V “The problems of out-of-school life of the pupil should 
receive primary consideration a part of the time.” 
1 Activities that will give rise to such problems as 

a School or home gardening involving 
(1) Keeping accounts of costs and profits 
(2) Earning money by labor 
(3) Scale drawing and planning 
(4) Bills 

b Store sales 
(1) Remnant sales 


ELEMENTARY ARITHMETIC SYLLABUS 79 


(2) Fractional discount sales (1/3 or 1/2 off) 
Have class bring in advertisements. | 
(3) Class sales of articles donated marked 1/2 or 
1/3 off 
c Household accounts 
(1) Keeping account of groceries and meat bought for 
a week 
(2) Cost of food, clothing, shelter, fuel etc. for each 
child. Find high, low and average cost (actual 
or approximate prices may be used.) 
(3) Which costs more, to clothe a boy or a girl? 
(4) School cost accounts 
Approximate cost a month for supplies for 
each individual for grade and possibly for school. 
Seek cooperation in endeavor to reduce cost of 
supplies for one month by economy in use of 
supplies for 1 month. 
Compute saving for 1 month for grade, for 
entire school. 
d Budget idea 
(1) Plan a personal budget for a child 
e Charts for records 
Health, weight, height involving averages 
f Thrift accounts 
(1) School bank accounts 
(2) City bank accounts 
g Keeping a record of speedometer on bicycle or auto- 
mobile so that child might know how many miles 
have been traveled from one point to another and 
also have an idea of mile and part of mile 


h Installing a radio set 
t Automobile trip account 


A Type Fifth Grade Project 
The following garden project was worked out by Miss M. West 
at Dunkirk, N. Y. It developed from a child’s actual school garden 
account. 
Aim: To teach how to keep an expense account 
Frank was in the fifth grade and could at last begin to make 
plans for a garden under the direction of a supervisor. 


SO THE UNIVERSITY OF THE STATE OF NEW YORK 


1 His father gave him a piece of land in a sunny place for his 
garden. Frank measured it and found it was 30 feet long 
and 15 feet wide. How many square feet of surface did he 
have to dig? 

What is another name for surface? 

2 After he had spaded it, March 30th, the little chickens began 
scratching in it. He went to the hardware store to get some 
wire to go around it. How much did he buy? 

What did it cost at 31/2 cents a foot? 

3 His father gave him fertilizer valued at $1. Frank wrote to 
his congressman at Washington for seeds, and received free 
on April 15th, flower seeds which were worth 75 cents. On 
May 30th he bought a dozen cabbage plants for 18 cents. 
On June 22d the supervisor advised him to get 1/2 pound 
of arsenate of lead for insects. It was 40 cents a pound. 

What were his total expenses? 

4 He began to keep an account with his garden. He did it as 

follows: 


Garden Account 


Debit Credit 


Cost of garden Rec’d from garden 


Mar. 30! To chicken wire....!$3.15 |;|May 15] 5 bunches radishes 


May 20] To tomato plants....| .25 ||May 30 at Scents see ee $0.25 
12 bunches radishes 
atiZ for i cents. <- .30 


5 Finish the debit side of his account. 


6 The last work was getting his vegetables and flowers ready 
for the school exhibit on September 19th. How many days 
did he have his garden? 
What do you think he could have planted as oo or earlier 
than March 30th, in your locality? 


7 In March he worked 2%4 hours; in April 10 hours and 10 
minutes ; in May 934 hours; in June 8 hours and 20 minutes; 
in July 5 hours; in August 2 hours and 15 minutes; in 
September 1 hour. What was his labor worth at 10 cents 
an hour? 

He said he would not charge this to his account because he 
thought he would get paid by developing his head, heart, 
hand and health, the motto of the junior project workers. 


10 


a 


12 


13 


14 


15 


ELEMENTARY ARITHMETIC SYLLABUS 81 


On May 15th he sold the grocer 25 cents worth of radishes. 
He put 6 in a bunch and received 5 cents a bunch for them. 
On May 30th he put the same number in a bunch but got 
only 5 cents for two bunches. The grocer paid him 30 cents 
for this lot. } 

How many radishes did he sell? 

How much did he receive for them? 


Write up his account for May. 
Notice which side of the account is larger. What does this 
mean? On which side did you put the money he received ? 


In June he sold a neighbor 15 heads of lettuce for the table 
at 5 cents a head, and 12 heads at 3 for 10 cents. 

What did he receive for his lettuce crop? 

With what do you think he replaced his lettuce and radish 
rows? 


Write up the account for June making up your own dates. 
Where is the balance now? 


His first tomato ripened on July 19th. He sold it for 10 cents 
because it was so early. On July 23d he sold 3 for 5 cents 
apiece; on July 24th, he sold 2 for 5 cents. On July 28th 
he sold % peck for 40 cents. On September 18th he sold 
YZ bushel of green tomatoes for 20 cents. 

Write up his account for his tomatoes. 

Has the balance changed sides yet? 


His flowers were his chief delight. He sold 3 bunches of 
sweet peas at 25 cents a bunch to a woman who would buy 
flowers every Saturday he had any to spare. He sold her 
3 dozen Comet Asters at 25 cents a dozen. He gathered his 
everlasting flowers when they were still buds and hung them 
up to dry. In September he sold 200 at 10 cents a dozen. 

What was his flower crop worth? 


Make up the dates and enter his flower crop in his garden 
account. 

Has the balance changed sides yet? 

What does this condition mean? 


At the school exhibit in September, he received a prize of 
50 cents for the best department garden record, a 50-cent 
prize for the best flower display, and 50 cents for the most 
perfect cabbage. 

He could not sell his cabbages because they were so plentiful, 
so he gave them to the chickens. 


82 THE UNIVERSITY OF THE STATE OF NEW YORK 


16 What did he gain on his garden project? 
Do you think his motto was a good one? 


For other activities see 20th Year Book of the National Education 
Association. A few are listed here. : 

Page 110. A ribbon sale for drill on fractions. 

Page 107. The play store or bank. 

Page 139. Learning thrift by keeping accounts. 


SIXTH GRADE 
The teacher should read carefully the general introduction to the 
syllabus. She should be familiar with the contents of the syllabus for 
the fifth year, and should fully understand the aims and objectives 
of the work for the end of the six-year period. 


Condensed Outline 


First Half 
A Review of integers 
B Common fractions reviewed with emphasis on fractional relation- 
ships 
C Decimals. More difficult work. Short processes 


Second Half 
A Bills with special emphasis on correct form 
B Personal accounts 
C Denominate numbers 
D Percentage 
I Finding the percentage 
II Finding what per cent one number is of another 
III Practical applications. Use first case of percentage 
1 Commission 
2 Profit and loss 
3 Simple interest (1 year only) 
4 Simple discount 


Expanded Outline 


Sixth Grade (First Half) 
A Integers 


I Drill for speed and accuracy 
1 Addition with a review of combinations 
2 Subtraction with attention to all difficulties 


ELEMENTARY ARITHMETIC SYLLABUS 83 


3 Multiplication with review of tables if needed 


4 Division with division tables including the forms 


1 12 
— of 12=—6 —=6 
2 2 
1 14 
—of 14=7 ——f 
Z 2 
1 16 
— of 16 = 38 a 
2 2 


5 Two and three-step problems with integers. Teach children 
to think out the more difficult problems in the steps of 
the solution, thus: 

a Given 

b To find 

c Estimate result (make an approximation) 
d Solution: Process 

e Proof: Verifying or checking result 


6 Reading and writing numbers to billions 

Relate to Geography in reading population, amount of 
national debts of various countries, values of exports and 
imports, etc. 

Note. For corrective work on the fundamentals use 
the Courtis Practice Drills, the Studebaker Drills, the 
Compact Efficiency Drills if available, or the Fassett 
Number Cards. They save time. Also consult second, 
third and fourth grade outlines for suggestions in grading 
the difficulties. 

For measuring the efficiency, accuracy and speed of a 
grade some standard test should be given 2 or 3 times a 
year. For reasoning, the Stone tests are good. Aim to 
keep the class up to standard. 


B Common fractions 


I Drill for speed and accuracy. It is suggested that each school 
system set its own standards for speed and accuracy in 
fractions. 


II It is also urged that some project or activity be undertaken 
by the class which will involve fractions. As each diffi- 
culty is encountered, stop for drill on the particular phase 
of the work. If the teaching of the fractions is done 
when some real need is felt for that specific piece of work, 


84 


Ww De 


A 


THE UNIVERSITY OF THE STATE OF NEW YORK 


the class will retain the acquired knowledge more clearly 
and will be much more interested in the work. A few 
suggestions for creating such arithmetical situations are: 
The cost of installing the radio 

Bringing in home accounts 

Making a collection of fractions from home problems, 

father’s business problems, cookbook problems etc. 

Imaginary or real store accounts 


III Place emphasis on 


1 


Sap fen: Wal Ss. fesy, (85) 


8 
9 


Addition 

Subtraction 

Multiplication 

Division | 

Reduction, using factoring method of L. C. M. 

Principles of fractions 

Fractional relations. The study of the whole and part 

should receive emphasis. This is the foundation of the 
work in percentage. (Review just before taking up 
percentage ) 

a Fractional part of a number, as 4% of 21 

b Fractional part one number is of another, as 2 is what 
part of 4° Make application here to school credits 
as 3 words missed out of 10 in spelling means what 
part missed ? 

c Find the whole when fractional part is given. Use 
small numbers and illustrate with diagram and 
otherwise. 

Problems from the text 

Problems from life 


C Decimals 
I Refer constantly to fifth grade outline, page 75. 
II Practice required in manipulating decimals up to and includ- 


ing six places. 


IIf The decimal is only a new form of writing the fraction (see 


1 


4 
3 


fifth grade course) 

Decimal notation 

a Much practice in reading as far as millionths 
b Drill in writing numbers 

Addition 

Subtraction 


ELEMENTARY ARITHMETIC SYLLABUS 85 


4 Reduction of decimals to fractions with a study of aliquot 
parts (see page 77) 
5 Multiplication 
For developing rule for pointing off see fifth grade 
outline. Short processes. 
6 Division of decimals 
Make a point in sixth grade of teaching the under- 
lying principles of pointing off. In the fifth grade the 
pointing off was taught as a mere mechanical process. 
7 Problems 


8 Short processes in division 


86 THE UNIVERSITY OF THE STATE OF NEW YORK 


Sixth Grade (Second Half) 
A Bills 
I Correct form 
1 Heading 
2 Ruling for purchases only 
3 Ruling for purchases and credit 
4 Dates 
5 Receipting 


Note. Bring to the class some bills from local busi- 
ness concerns. Study different forms accepted by busi- 
ness houses. 


Headings 


Harris Anderson & Co. 


Des Moines, lowa 
Sold to Mrs Raymond Holmes 


67 Avenue A 
Des Moines 
September 5, 1922 
Or 


New York City, June 12, 1921 
Messrs. Raymond & Co. 


Syracuse, N. Y. 
Bought of Jones, Tompson & Co. 


Ruling: Purchases Only 


3 Ibs. coffee @ $.54 $1.62 
etc, 


Received payment 
Signed by firm name 


Ruling: Purchases and Credits 


10 lbs. coffee @ $.54 $5.40 
20 lbs. tea @ $.60 $12.00 
Cash $7 .00 
$17.40 
Balance $10.40 
Paid 


Signed by firm 


ELEMENTARY ARITHMETIC SYLLABUS 87 


II A simple form for receipt by principal and by agent. Bring 
printed receipts to class. 


B Keep a simple personal expense account 
I Debit, credit and balance 
Assume you have an allowance of $5 a month; required 
to buy school supplies, personal supplies, etc. 


II Teach budget idea. Figure out a budget for a school account 
or a boy’s or girl’s personal account. Show how the budget 
may be used as an aid to thrift. 


C Denominate numbers 


I Complete all denominate number tables that have not been 
completed. 
ie winear 
2 Avoirdupois 
3 Liquid 
4 Dry 
5 Time 
6 (Optional) Square measure 
a Square inch, square foot, square rod, acre 
b Areas of rectangles 
Note... Square measure may be taught if a need is felt for 
it. Possibly some project may call for it, otherwise it may 
be left until later. 


Possible Method of Procedure 
Square measure (if taught) 

1 Pupils bring to class actual measurements used in their 
experiences of buying or making articles. Domestic 
science and manual training will give practical 
measurements. 

2 Observe nearby lots; tell why they are attractive. Bring 
in a point of civic pride. 

3 Study a particular lot, possibly the school lot. 

a Measure the lot 
b Measure the building 

Note 1. Show a need for changing from one denom- 
ination to another. 

Note 2. Draw the lot to a scale. Show how to find 
the surface expressed in square units by so many rows 
of a certain number of square units as 6 rows of 5 
square feet. 


88 THE UNIVERSITY OF THE STATE OF NEW YORK 


Note 3. We may find the proportional relationship 
of building lot. The walk may be laid about the part of 
the lot where it is needed. 


II Reduction of denominate numbers 
1 Change from lower denominations by division. Two-step 
change only — 3 denominations 
2 Change to lower denominations by multiplication. Two- 
step change — 3 denominations 
III Addition and subtraction of denominate numbers 
1 Two denominations only 
IV (Optional) Multiplication and division of denominate 
numbers 
Note. If deemed necessary, this may be taught. The 
committee recommends omission. 
V Problems: original problems by pupils 
For suggestion activities see end of sixth grade outline. 


D Percentage 


I Review fractional relationships and aliquot parts 
1 Fraction decimal and per cent equivalent 
Fraction Decimal Rene cent 


4 


33 


75% 
II Finding percentage 
1 Have pupils find the 1/100 part of numbers 
2 Lead pupils to see that we find 1/100 part by multiplication 
3 Teach that to find the percentage of a number you multiply 
the number by per cent 
4 Many practical problems 


III Finding one number is what per cent of another. Use simple 
numbers, relate to fraction parts and apply to problems 
that are practical as: A child has 8 words correct out 
of 10 words, what per cent are correct? 


100 
IV The whole amount of anything is 100 per cent of it or — 


100 
V Applications of percentage 


1 Commission 


a Commodities bought or sold for another at a given rate 
per cent 


ELEMENTARY ARITHMETIC SYLLABUS 89 


b ‘Terms that are necessary, as agent, commission, amount 
of sales etc. Find commission. 


2 Profit and loss 
a Gain or loss and selling price at a given rate of gain 
or loss 
b ‘Terms that are commonly used. Find the gain or loss 
when the cost and rate of gain or loss are known. 


3 Simple interest 
Use time exactly, 1 year 


4 Simple discount 
10 per cent off etc. 
Find discount 


Arithmetical Activities 


The following arithmetic problem work is suggestive of what can 
be done with a schoolroom situation. It is furnished by Miss Hession 
of Dunkirk and was worked out in a sixth grade. It proved much 
more interesting to the class than the ordinary text problem besides 
furnishing practical work in liquid measure, in reduction from half 
pints to quarts and gallons, decimals, fractions, and United States 
money simple account keeping etc. 

Milk was furnished to the children of the school and the following 
data was used for arithmetic. 


Monday Kindergarten ordered 14 half pint bottles 
Monday Grade I ordered 25 half pint bottles 
Monday Grade III ordered 13 half pint bottles 
_ Monday Grade IV ordered 15 half pint bottles 
Monday Grade V ordered 10 half pint bottles 
Monday Grade VI ordered 5 half pint bottles 


Question: 1 How many bottles? 
2 How many quarts? 
3 How many gallons? 
4 Tf children pay 3 cents per bottle and the milkman gets 10 cents 
per quart, how much money is left? 
5 How many pounds of graham wafers can be bought at 12% 
cents per pound with the money left? 
6 How many boxes of straws at 23%4 cents per box? 
7 How. long will it take to pay for a 10-pound box of graham 
wafers at 121% cents per pound? 


90 THE UNIVERSITY OF THE STATE OF NEW YORK 


8 How long for 3 boxes of straws at 231% cents per box? 

9 How long for one 10-pound box of wafers at 12% cents per 
pound and 1 box of straws at 2314 cents per box? 

Note. These problems are not all given at one lesson. Problem 1 
may be dealt upon for at least a week as the supply changes each day. 

Then problem 2 may be taken up etc. An account of the total 
supply for the week, month etc. may also be kept. 


Other Activities Possible 
1 Collecting and making problems from business, from home etc. 
2 Keeping accounts 
Garden accounts, see fifth grade 
Earning and saving money 
3 Drawing to a scale 
4 Playing bank for making out check, deposit slip, saving account 
and simple interest 


Information Pertinent to Sixth Grade 


1 Knowledge of business forms: bills, receipts 

2 How to keep a simple personal expense account, debit, credit 
and balance 

3 Idea of a budget 

4 Information pertaining to household costs, especially of food, 
shelter, fuel, clothing 

5 Farm arithmetic 


ELEMENTARY ARITHMETIC SYLLABUS 91 


INTRODUCTION TO ARITHMETIC IN UPPER GRADES 


In the work in arithmetic during the first 6 years, stress has been 
laid on the development of skill in the fundamental operations. 
There has been considerable emphasis on the mechanical manipula- 
tion of numbers. The child should have become proficient in 
handling whole and mixed numbers in fundamental processes. He 
should have gained a knowledge of simple denominate numbers. He 
should have a working knowledge of the decimal system and working 
ability in the simple relationships involving arithmetical processes. 
Problems have been given from time to time so that he can grasp 
a simple situation and from it lead to a conclusion. 

The point of view changes somewhat with the beginning of the 
work in the seventh year. Pupils are entering upon the early 
adolescent period and are gradually developing a social consciousness. 
The “why” of things begins to mean more to them. They begin, 
although dimly, to sense something of the great social organism of 
which they are a part. The time is coming earlier or later, very 
early with a large majority, when they will be compelled to take 
their places in the world. They begin to develop an interest in what 
it is all about. The school must recognize this psychological change 
in the pupils. The work in arithmetic, therefore, as in other sub- 
jects, in the higher grades must be approached from a different 
point of view, and it has at the same time a somewhat different aim. 

While the work is organized by half years for both seventh and 
eighth grades, it is so organized merely for grouping purposes. It 
is appreciated that the subject must now be handled in larger units 
of study and that the particular distribution of topics and arrange- 
ment of material will depend in large measure on the school organiza- 
tion of the individual community. The work for these years should 
therefore be interpreted largely as a topical arrangement rather than 
as necessary to be included in half years. The grouping should be 
determined in terms of local needs. 

In the seventh year the solution of problems should be greatly 
emphasized. Speed and accuracy are still important factors in the 
work, but the development of the power to reason, at the period of 
life when this phase of mental growth begins to manifest itself 
prominently, is the great aim. At this period of his school life the 
pupil’s ability will be determined in part by the special courses which 
he may be pursuing in the intermediate school or in the junior high 
sckocl where such an organization obtains. 


92 THE UNIVERSITY OF THE STATE OF NEW YORK 


It is fundamentally important that the teacher fully appreciate the 
purpose in curriculum modifications at this point. Problems must 
be interesting. They must bear on the pupil’s experiences in the 
shop or the kitchen, on the playground or elsewhere. They must be 
of a variety that will lead the pupil to reflect, to investigate and to 
feel that he himself is concerned in the social and economic environ- 
ment of which he is the center, and further that the problems that 
_are presented to him come out of a living world in which he is to 
take his place. 

Here are to be emphasized the principles of business, of invest- 
ment and of thrift, and the situations which are to make for honesty 
and good citizenship. The character of the work in this grade may 
determine largely the strength or weakness of the pupil in his future 
work in mathematics. 


ELEMENTARY ARITHMETIC SYLLABUS 93 


SEVENTH GRADE 


The teacher should read carefully the general introduction to the 
syllabus and should be familiar with the development of the subject 
as given in the syllabus and especially with the work of the sixth 
year, the aims attained at that point, and the new approach to the 
subject in the higher grades. 


Condensed Outline 
First Half 


Drill on fundamentals 
Review of fractions and decimals 
Percentage 
Applications of percentage 
I Profit and loss 
II Commission 
III Commercial discount 
IV Insurance 
V Taxes 


GUO pS 


Second Half 


Interest 

Notes 

Bank discount 
Banking 
Measurements 


,HOOWS 


I Review of tables 
II Longitude and time 
III Linear measure 
IV Areas — square, triangle, rectangle, parallelogram, circle 
V Board measure 
F Construction to scale. Map scales 
G Miscellaneous work on problems in percentage and its applica-. 
tions, also on measurements 


Expanded Outline 
Seventh Grade (First Half) 


A Keep up drill on fundamentals with’ speed tests. Special attention 
should be devoted to pupils who are below standard. Graphs 
drawn by the pupil may be used to keep his scores, either for. 
purposes of comparisons of medians with those of other classes 
or with those of other individuals. 


04 THE UNIVERSITY OF THE STATE OF NEW YORK 


B Review work in fractions, including decimals. Drill on expressing 
numbers as fractions, decimals, and per cents. 

C Reteach finding per cent of a number. Lead pupils to see that 
since in finding per cent of a number, the number is multiplied 
by the per cent, therefore the per cent of a number or per- 
centage is the product of that number and the rate, another 
name for per cent. 

Number times rate per cent = percentage 
If the per cent of a number or percentage is a product of said 
number and the rate, either factor can be found by dividing 
the product by the other factor. 


I Examples 
1 Bie of 400=? 
— of 400 = 24 
100 
2 24 is 6% of what number? 
24 — .06 = 400 
or 
6 100 
24 — —_— = 24 x —-= 400 
100 6 
3 24 is what per cent of 400 
24 6 
24 — 400 = .06 or —-=—-=6% 
100 100 


Note. Many prefer the following presentation: 

1 To find the per cent one number is of another: In a certain 
school of 600 pupils, 240 are boys, what per cent of the 
pupils are boys? 

240 
240 boys are —— of 600 pupils 
600 


240 2 
—  =- or .40 or 40 per cent 
600 
2 A given number is a certain per cent of a number, what 
is the entire number? For example, in a school where 
40 per cent are boys, there are 240 boys, how many pupils 
are there in the school? 


40 2 
40% = — or — 
100 5 
2/5 of the pupils enrolled = 240 pupils 


ELEMENTARY ARITHMETIC SYLLABUS 95 


1/5 of the pupils enrolled = 1/2 of 240 pupils or 120 
pupils 
5/5 of the pupils = 5 & 120 pupils or 600 pupils 
Note. Some teachers will prefer not to change their 
per cents to fractions. In that case the last solution 
might be given as follows: 
40% of the pupils = 240 pupils 
1% of the pupils = 1/40 of 240 pupils = 6 pupils 
100% of the pupils = 100 X 6 pupils = 600 pupils 
Note. The pupil should know thoroughly the fol- 
lowing cases of percentage: 
1 How to find a per cent of a number 
2 How to find of what number a given number is a 
certain per cent 
3 How to find what per cent one number is of another 


D Much drill on these processes as abstract work 


FE Drill in problem work involving these three cases: 
1 Use problems that come in the child’s life as: 
Attendance at school 
Amount of work finished 
Per cents earned 
Games 


F Complete profit and loss 

Cost is sum paid for article 
Per cent of cost is gain or loss when article is sold 

Note. Here discuss why a merchant should receive a gain; 
what is considered a legitimate profit; why this profit may be 
greater in one place than in another. Show the relation of 
turnover to profit; the meaning of overhead; the reason for 
marked-down sales; the value of good salesmanship. 


I Problems 
1 To find gain or loss: I buy a house for $8000 at a gain of 
20%. What is the gain? 


$8000 
.20 or 1/5 of $8000 = $1600 


$1600.00 
2 To find selling price in above problem: 


96 THE UNIVERSITY OF THE STATE OF NEW YORK 


Note. Either of the following solutions is good: 


100% represents the cost or 5/5 
20% represents the gain or 1/5 


120% represents the selling price or 6/5 


or 6/5 of $8000 = $9600 


$9600.00 selling price or $8000 cost 
1600 gain 


$9600 selling price 


3 If the above house were sold at a loss of 20% what was the 
selling price? 


100% represents the cost or 5/5 
20% of the cost represents the loss or 1/5 
80% of the cost represents the selling price or 4/5 

$8000 

.80 

$6400 selling price or 4/5 of $8000 = $6400 

or 

$8000 cost 
1600 loss 


$6400 selling price 


4 If the sale of the above house at a loss of 20% involved a 
loss of $1600, what was the cost? 
20% of the cost = $1600, the loss 


.20) $1600 1/5 of the cost = $1600 
$8000 
20) $160000 5/5 of the cost = 5 & $1600 = $8000 
5 If a man gained $1600 on a house that cost $8000, what per 
cent did he gain? 
20 1600 
8000)1600 or $1600 is 
8000 


of $8000 


1600 
-20 = 20% — = 1/5 or 20% 
8000 


ELEMENTARY ARITHMETIC SYLLABUS 97 


G Commission reviewed and new cases added 
I Definition: a term used in the commercial world to designate 
the name of money paid to an individual for transacting 
business for another. 
Review of definitions of terms taught in sixth year. 


II Reteach amount of sales & rate of commission = commission 


1 When total cost is required, the commission and any other 
expenses connected with the purchase must be added to 
sales price. This is when buying. 


Z When profit from sale by a commission merchant is 
required 
The commission and any other expenses incident to the 
sale are added. This sum subtracted from sales 
price = proceeds 


III Facts about which the pupil should be positive 

1 Amount of sales times rate of commission equals com- 
mission 

2 Amount of sales plus commission equals total cost 

3 Amount of sales minus commission equals sum remitted 
to principal 

4 Amount of sales equals commission divided by rate of 

commission 


5 Rate of commission equals commission divided by sales 
Notre. Any good form should be accepted. See types 
under profit and loss. 
Label work so thought is clearly expressed. 
The multiplier should be an abstract number. 
Much drill in real problems which are furnished by 
teacher, also by pupils. 


‘H Commercial discount 
I Terms used: list price, rate of discount, net price 


II Why discounts are given and that more than one discount is 
frequently given 


III List price times rate equals discount 


IV Discount is a per cent of a number 
An article listed at $250 is sold at a discount of 25 per 
cent. How much is paid for this? 


98 THE UNIVERSITY OF THE STATE OF NEW YORK 


100% = marked (list) 
25% = rate of discount 


75% (net) 
$250 list price 
0, 
woos $750 
1250 or 3/4 of $250 = —— = $187.50 net price 
1750 . + 


$187.50 net price 
V List price times (100% minus rate of discount) equals net 
price 
VI $250.00 marked price or list price 
Ray Arte) 6, 


$62.50 discount 

Therefore, list price minus net price equals discount 

An article for which I paid $187.50 was sold at a discount ~ 
of 25 per cent. What was the original price? If at a dis- 


count of 25 per cent, it was sold for 3/4 of original price, 
$187.50 


3 , 4 
$187.50 — — = 62.50 * — = $250 
4 * 


For other forms of solution see pages 93-94. 
If an article sells for $187.50 at a discount of $62.50, 
what is the rate of discount? 
$62 250 = $187750°)25%en 1/4" 0r8Z5% 
VII To find the net price when a series of discounts 
Goods marked at $125 have discounts of 10%, 5% and 2%. 
Find the price received by merchant. 
100% 100% 100% 
10% 5% 2% 


.90 OS .98 
$125 list price $112.50 new list price $106.88 new list price 
.90 95 .98 


_—_ 


$112.50 net price 56250 85504 
101250 96192 
$106 .8750 $104.74 net price 
$125.00 
104.74 


$20.26 discount 


ELEMENTARY ARITHMETIC SYLLABUS 99 


VIII Bills when discounts are given 
Form 
Schenectady, N. Y., January 3, 1923 
Teed, ormtith or CO. 
Sold to D. S. Black 
Terms 10%, 5% 


~ 


500 yds. silk @ $2.50 $1250.00 
Less 10%, 5% 181.25 
$1068.75 


Received payment 


Jat. Siiths we Co. 
I Insurance 
I Definition of terms used 
II Kinds of insurance. Why a protection? 
III Growth of insurance 
Is it profitable? For whom? Why? 
IV Essentials to be found in policy 
1 Study of a real policy 
a The sum for which insured 
b The agreement clearly stated of company to insured 
party 
c Term of premium 
Note. Investigation of rate of insurance compared with 
neighboring places, also as compared with rural sections. 
What factors determine the rate of insurance? 
V Lead pupils to see this is an application of percentage and 
solved in same way, therefore 
1 Face of policy times rate equals premium 
2 Premium divided by rate equals face of policy 
3 Premium divided by face of policy equals rate 
J Taxes 


I Show why taxes are necessary, how all must share. Lead 
pupils to feel that taxes should not be avoided or neces- 
sarily lowered, but that the public should receive adequate 
returns for these taxes. 

II Talk about budget. Why is it good? 

III Kinds of taxes: city, county, state, federal; also personal on 
property as well as income. When levied on imported 
commodities a tax is known as duty. 

IV Bonding individuals for tax collectors. Why? Does this 
build character? Study the form. 


100 THE UNIVERSITY OF THE STATE OF NEW YORK 


V_ Assessors 


VI Facts to be taught. How to find. 
1 Rate of tax. Tax to be raised divided by total valuation 
2 Individual tax on certain piece of property. Valuation of 
property times rate of tax 
3 Total tax to individual when collector’s fee is included 
a Tax times rate of fee equals collector’s fee 
b Tax plus collector’s fee equals total tax 
Note. Use the common tax of your own community. 
Teach common method of expressing rate; dollars on a 
thousand. Compare this rate with neighboring communi- 
ties and see why they differ.. Decide whether the cause 
justifies this difference. 


Seventh Grade (Second Half) 
A Interest 
I Terms used: principal, or 1f note face of note; rate of 
interest; interest; and amount 
Interest is very similar to rent. Rent is pay for the use 
of property at given rate for certain time, while interest 
is pay for the use of money at a given rate for a certain 
time. 


II Therefore, principal times rate times time equals interest 


III Time is the troublesome factor because it may be expressed 
in three units of measure 
Have a talk with the pupils about the various ways which 
people have used to compute interest but lead them always 
to see it is the product of the three factors above mentioned. 
Allow any standard way of computing interest 
Cancellation method and 6 per cent method are recom- 
mended 
For short time loans the bankers’ method is desirable 


IV (Optional) Show how the converses may be solved. 
Interest divided by principal times rate equals time 
Interest divided by time times rate equals principal 
Interest divided by principal times time equals rate 
Much time is spent in finding interest. Accuracy and 
speed is important in this. Problems in which interest is 
employed should be given. Little time is spent with the con- 
verses and their application. 


ELEMENTARY ARITHMETIC SYLLABUS 101 


B Notes 


I Definition: a written promise to pay a certain sum of money 
to a certain party at a certain time with certain rate of 
interest or without interest 

II Essentials of a note 
1 Parties 
2 Negotiable or nonnegotiable 
3 Indorsements. Selling of notes 
The pupils should be able to write any form of note and 
to know the names of the parties to a note. 
C Bank discount 
I A note is like a commodity for sale 
II Banks being dealers in money are the places to sell or buy 
notes 
III Term of discount. Pupil should see why there are two dates 
IV How the interest bearing note is affected 
V Discount on an interest bearing note. Pupil should under- 
stand why the new face amount is sum discounted 
VI Terms the pupil should be taught: bank discount, proceeds, 
new face, term of discount, date of maturity 
D Banking (Jn this grade only slightly studied) 

I Different kinds of banks 
1 Savings 
2 National 
3 Private 

II How to deposit money 

III How to withdraw funds 
IV Checks 

1 Why good 

2 How to write 

3 Certified 

V Drafts 
1 How they differ from checks 
2 Why used 

VI Other good ways of sending money 
E Measurements 

I Review all tables already learned and teach those your school 

requires and not already taught 

II Review all work in denominate numbers so pupils will reduce 

answers to the terms in general use 


102 THE UNIVERSITY OF THE STATE OF NEW YORK 


III Longitude and time 

1 Teach only enough to find practical change in time and 
how the longitude of the places causes this change. 

2 Many schools may feel that this is a nonessential in 
arithmetic, a sufficient amount of this topic being taught 
in geography. 

3 When it is taught use following steps: 

a To find the difference of longitude. If places are in 
same direction of longitude from the prime meridian, 
subtract; if in different, add. This gives difference 
of longitude 

b Divide difference of longitude by 15 to obtain difference 
of time 

Note. Show how this 15 is obtained. 
c Having difference of time, show how to get the time of 
a place 
(1) If traveling eastward, add 
(2) If traveling westward, subtract 


IV Review linear measure. Teach meaning of construction of 
straight line, circumference and perpendicular 


V Teach names of type forms of following surfaces: square, 

triangle, rectangle, parallelogram, circle 

If table of square measure has not been taught, introduce 
it here. | | 

Show when areas are used. Be sure pupils see the surfaces 
of various solids. 

Spend much time that pupil may see that circumference of 
circle is linear measure while the surface is square measure. 


VI How to find areas of surfaces named above. Make study in 
and about building to find these 


VII Board measure 
1 A board foot is a piece of timber 1 foot long, 1 foot wide 
and 1 inch or less thick. Use cancellation method 
2 Number of pieces times length in feet times width in feet 

times thickness in inches equals number of board feet 
3 When price is required, include it in above formula. For 
example, find the cost of 18 pieces of timber 16 feet 
long 18 inches wide and 1%4 inches thick at $30 per M. 
Lae) 30 1944 
18 xX 16&*& — xX — X —= 
LAZY 1000"? TOO 


or $19.44 


ELEMENTARY ARITHMETIC SYLLABUS 103 


For schools that take up the junior high school idea or the 4-3-3 
plan instead of the older 8-4 plan, the following is suggested. It 
may be introduced in other places during the year rather than placed 
at any one step. 


F Geometry 
I Approach this subject by teaching pupil by means of well- 
planned observation and experiment the most important 
geometric forms which are found in our everyday life. 


II Teach how to construct these geometrical figures 

1 Use T-square, triangle, protractor and compasses 

2 Construction of a perpendicular bisector of lines, polygons 
which the pupil has found in practical life and forms 
found in nature 

3 Direct measurements drawn to a scale 

From these compute some capacities and thus lead 

pupils to see when we demand absolute and very accurate 
measurements; when we may use approximations. 


4 Indirect measurements as measuring heights etc. 


G Review all applications of percentage by means of miscellaneous 
problems. Use problems that are real but are sufficiently simple 
to enable pupils to feel they are masters of the situations and 
thus keep up their interest. 

Lead pupils to bring in problems and stimulate them to give 
problems which will test the ability of their classmates. Fre- 
quently take some of their problems for your tests. 

Lead pupils to see the three forms of percentage. Let the 
problems also include banking, bank discount and measure- 
ments. 


H (Optional) Situations for seventh grade work 
I A boy is given $100. How shall he invest it so that he may 
realize the more for his use when ready to enter college? 
II Study of wills. How gifts may be made to mean the most 
to the beneficiary 
III Build a radio set 
IV The parcel post 


104 THE UNIVERSITY OF THE STATE OF NEW YORK 


EIGHTH GRADE 


The teacher should be familiar with the development of the work 
during the previous years, should study very carefully the syllabus 
for the seventh year, and should keep clearly in mind the aims of the 
course in terms of the larger social interests of the pupils of early 
adolescent age. 


Condensed Outline 
First Half 


A. Review fundamentals. Check proficiency of the class by means 
of standard tests 


B Measurements 

I Reviewed 

II Cubic measure, volumes 

III (Optional) Metric system 

C Ratio and proportion 
D Powers and roots; square root and its applications 
E General review to percentage 
F Review percentage and its applications — studied more extensively 

‘than in the seventh year 
G Banking practice 
H Stocks and bonds 
I Investments 
J Review completed including new work of the grade 


Second Half 
A Drawing to scale 
B Lines and angles 
C Construction 
D Graphs 
Lebar 
II Circle 
III Broken line 
IV (Optional) Pictographs 
E Introduction to simplest literal symbols and expressions 
I Meaning 
II Definitions 
III Reading and writing from dictation simple literal expressions 
Note. It is recommended that the consideration of nega- 
tive quantities and the work in subtraction be deferred until 
‘the ninth year. 


ELEMENTARY ARITHMETIC SYLLABUS 105 


F Addition of literal terms (positive results only) 
G Numerical substitution 

H Certain axioms 

I The equation 

J The problem 

K The formula 


Expanded Outline 
HKighth Grade (First Half) 
A Review fundamentals 


Tests for speed and accuracy, include work in fractions, 
decimals, percentage and interest. 

Keep this work simple enough to retain interest because of 
success gained but sufficiently advanced to produce a real 
development of power. 

Use aliquot parts in the above whenever an advantage. Gain 
skill in multiplication by extending their use, as: 125 & 7247 
(consider 125 as ¥% of 1000) add three ciphers to 7247 and 
divide by 8. Emphasize short processes of multiplication and 
division. 

Demand accuracy and speed. 

B Measurements 


I Work of previous grades 
II Where volume is required 
III Table of cubic measure. Teach 231 cubic inches in gallon 
(2150.42 cubic inches in bushel rather than 1%4 cubic feet) 
IV How to find volume of parallelopiped, prisms, cylinder 

Lead pupils to see that they must think of the shape of 
the base before they can compute volume. 

Be sure pupils know when to find surface, when volume ; 
also that surface means square units, two dimensions — 
volume means cubic units, three dimensions. 

Spend considerable time on circles and cylindrical type 
forms. 

V Give time for some real problem of your community in 
measure work as: constructing a road, laying out a park, 
constructing a public building, decorating a set of rooms 

There may not be time to do all of this but in a general 
way pupils should be encouraged to familiarize themselves 
with costs of these things. 

VI (Optional) Metric system 


106 THE UNIVERSITY OF THE STATE OF NEW YORK 


C Ratio and proportion 

I Lead pupils to see that we often express measures by com- 
parison as size of buildings, fields, height of hills, trees 
etc. Then show them that we can also express a com- 
parison of number. Division is really such a comparison. 
Fraction is an expressed division. It is a ratio. An 
equality of ratios gives a proportion. 

II The pupils now know definition of ratio and proportion. 
Teach other terms used: means, extremes, antecedent, 
consequent, also symbols used, : and :: or = 

III Product of extremes equals product of means 

IV How to solve by cancellation so as to find the missing term 

V How to state a proportion 

VI How to divide a number according to given ratios 

1 Add the terms of the ratio 

2 Form fractions, using the sum of the terms as denom- 
inator and each term in order as numerator 

3. Multiply the number by these fractions 

Show pupils how proportion may be employed in 

many problems, always when a comparison of similar 
conditions exists. Give real problems for drill in ratio 
and proportion. 


D Powers and roots 
I To find the area of a circle, the radius squared was used. 

From this teach the pupil that he used the second power 
which is a product obtained by multiplying a number by 
itself. The third power may be found by using the same 
factor 3 times and the fourth power by using a number 
as a factor 4 times, etc. The power may be known from 
the number of times the number is used as a factor. 

II The power is indicated by a little figure written above and 
to'the*right*of'a’ nuniber*asi2> ee Zee cee 
Cle. 

III This little figure is called the exponent and always shows 
how many times the number is taken as a factor. 

1 Give drill in raising numbers to different powers. Use 
whole numbers, fractions and decimals. This gives 
opportunity to acquire speed and accuracy. 

2 Be sure that the pupil knows that the second power of a 
number is the square. (He should also know that the 
third power is the cube.) To find the area of a square, 
square the length of the side. 


ELEMENTARY ARITHMETIC SYLLABUS 107 


IV The number repeated as a factor to produce the power is a 
root. A root of a number, then, is one of the equal factors 
used to produce the number. The square root of a number 
is one of its two equal factors. 

V Practice in finding roots by means of factoring 

VI Square root 


The explanation of square root may be based on a 
geometrical figure. Some teachers may prefer to make the 
teaching of square root a purely mechanical process, 
deferring the explanation until the topic is studied in 
algebra. 


VII How to find hypotenus of a right triangle when the legs are 
known and the length of one leg when the hypotenuse 
and other leg are known. The diagonal of a rectangle; 
solution of problems involving square root. 


E General review as far as percentage 


I Notation and numeration 
TT Fundamental operations 
III Fractions 
1 Definitions and principles 
2 Operations 
3 Fractional relationships 
4 Miscellaneous problems 
IV Decimals 
I Reading and writing 
2 Operations — special emphasis on division 
3 Aliquot parts 
4 Short processes (much drill) 
V Denominate numbers 
F Review of percentage 
I The three relationships, problems, applications of percentage 
In the review of applications of percentage it is not meant 
that the problems shall be difficult or that the. study of the 
various topics shall be exhaustive. It is, however, advised 
that the teacher shall send his pupils afield for material that 
not only will motivate the work but will furnish a fund of 
useful and interesting information. This will be valuable 
in later life and may not be made available to the pupil in 
any other wav. 
It is suggested that interest and enthusiasm may be stimu- 
lated by the organization of members of the class into a 


108 ' THE UNIVERSITY OF THE STATE OF NEW YORK 


Wide World Club. Regular meetings may be held and local 
business and professional men may be invited in to speak. 


Suggestions for talks: 


Banker, banking practice, bank loans and discounts, 
investing money 
Village, city or other official, taxes and the tax budget 
Insurance agent, fire insurance, life insurance 
Merchant, the overhead, the turnover, merchandising 
Traveling salesman, salesmanship, different methods of 
selling: on salary, commission, bonus; prevailing 
commissions allowed 
County, village or city treasurer, the work of the treas- 
urer and the collection of taxes 
President of the board of education, financing the school 
Street commissioner or superintendent of highways, 
road construction, kinds and cost; concrete as a 
building material; its value and cost; keeping the 
streets clean 
Postmaster, the business of the local post office 
Contractor, building construction, steel, brick, stone 
Water commissioner or superintendent, the local service; 
its problems and cost; water bonds as an investment; 
municipal ownership 
Advertising agent, cost and value of advertising, rates, 
keyed ads 
Many other classes of speakers will occur to the wide- 
awake teacher. Speakers should be urged to tell about the 
arithmetic of their business, but this should not drift into 
dry statistics. Pupils should be encouraged to take notes 
and from notes construct problems based on conditions 
they have heard described. They should be encouraged 
to ask questions and discuss the talks they have heard. 
1 Commercial discount 
a Same as in seventh year 
b Emphasize bills, collect all forms that pupils can find. 
Study advantages 
c Overhead charges 
d Advertising. Have pupils collect and bring to class adver- 
tisements announcing discounts and marked-down sales 
e Salesmanship. Study factors involved in good salesman- 
ship. The commercial traveler. Why necessary 


ELEMENTARY ARITHMETIC SYLLABUS 109 


2 Profit and loss 
a Marking goods so as to allow discounts and still make a 
profit 
b Purchasing goods at a discount for cash; advantages 
c Legitimate profits, variations in profits, in different lines 
of business 
d The turnover 
1 Its meaning 
2 Its desirability and advantage 
e Including interest in sale price when money is borrowed 
for purchasing goods 
f Multiple discounts 


3 Insurance 

a Kinds: fire, life, accident, health, burglary, liability etc. 

b Principle involved: Many parties cooperating by paying 
comparatively small sums for a longer or shorter period 
can make certain the payment of losses in the relatively 
few cases when they occur. The law of average. 

'c¢ A protection against loss 


¢e@ d Investment of insurance funds. State supervision of in- 
surance companies 


e Have pupils discover names of prominent insurance com- 
panies. Encourage them to discover local rates of fire 
insurance and find out factors that may reduce those 
rates. Let the class examine policies covering fire, 
accident or life insurance 


f Discuss the advantages of carrying insurance and the 
danger of neglecting to do so 

g Have a local fire insurance agent talk on fire hazards, 
board of underwriters, local fire insurance rates for 
various kinds of risks, (whether high or low and why) 
the value of a good fire department, etc. 

h Have a life insurance agent give a simple talk on the value 
of life insurance, two or three common forms of policy 
and the cost 


4 Commission 
a Same as in the seventh year but more difficult problems. 
Combine it with other items of expense to arrive at 
total cost 


110 THE UNIVERSITY OF THE STATE OF NEW YORK 


b Encourage pupils to discover common articles of mer- 
chandise handled by commission agents and varying 
rates of commission charged. Factors that make these 
differences 

Saaaces 

a Take up as in seventh grade 

b Reason for taxes 

c Secure a local budget. Find the total assessment. Make 
problems from the local conditions 

d Effect of taxes on rent. Who really pays the taxes? The 
effect of high taxes on industry and the location of 
industry 

e General provisions of the income tax, exemptions, state 
and national. Simple problems 

f Tax-paying and good citizenship 

g Kinds of property exempt from taxes. Why? 

h Have a talk from some village, city or county officer on 
the tax question and the making up of the budget 


6 Interest and bank discount 

a Treat the subject as in the seventh year. Introduce 
partial payments on a note. Use not more tHe two 
simple payments 

b (Optional) The Federal Reserve Bank and the relation 
of the members of the system to the Federal Reserve 
Bank. Rediscount 

c Simple savings bank problems involving compound interest. 
Use 4 per cent only. 

d See if pupils can find out how interest varies in different 
banks 

e Secure forms of note. Discover the method of discounting 
at. a local bank. The meaning of protest. The fee 
charged 


G Banking practice 
I Kinds of banks (see page 100). Trust companies. The 
broader powers of the latter. Bank examiners 
II Two reasons for banks 
1 To keep money safely 
2 To lend money needed for business and industry 
III Security for loans. Collateral. Find out different kinds 
~IV Checking accounts 


ELEMENTARY ARITHMETIC SYLLABUS Lut 


V Interest accounts 
VI Checks, certified checks, drafts, indorsements 
VII Certificates of deposit. Bank book. Monthly bank state- 
ments to depositors 
VIII Penalty for writing checks on bank not containing the 
drawer’s account. Seriousness of drawing a check for 
more than the amount of the deposit. Reconciling balance 
each month 
IX. Safe deposit boxes 
X The class should visit a bank and be permitted to see the 
vault, safe deposit boxes, etc. and have as much explained 
to them as they are able to comprehend 


H_ Stocks and bonds 


I Terms and definitions: Stocks, bonds, par, premium, 
quoted price, brokerage, discount, preferred, common etc. 
Note. Help pupil to see difference between a share of 
stock and a bond. If possible show class a stock certifi- 
cate or a liberty or other bond. Comparative advantages 
of owning stocks or bonds. 
II (Quoted price plus brokerage) times number of bonds 
equals cost 
III (Quoted price plus brokerage) times number of shares 
equals proceeds 
IV Difference between cost and net selling price equals gain 
or loss. Periods of depression (Optional) 
V Dividend. It is always the par value (usually $100) times 
rate of dividend. This is often expressed thus: U. S. 4s, 
meaning 4 per cent dividend. 
VI Rate of income. Divide the dividend as computed above 
by the cost of share 
VII Have pupils trace fluctuations of some bond for a week or 
more and report 
VIII Meaning of buying on margin; its danger 
Note. The pupil should be made to see that it is not 
easy for an inexperienced person to know when to invest 
to the best advantage. Competent advice should always be 
sought. Emphasize. 


I Investments 
I Kinds 
1 Speculative or uncertain 
2 Conservative, usually safe 


Lis THE UNIVERSITY OF THE STATE OF NEW YORK 


II Principles 

1 Absolutely safe investments bring lower rate of interest 

2 Long term investments usually bring lower rate of income 

3 Promises of high rates of interest usually signify doubtful 
security 

4 Danger of investing money without obtaining information 
from some disinterested authority, one who is in a posi- 
tion to know fully the safety of the investment 

Note. Pupils should be encouraged to find from banks 
or other sources information concerning the reliability of 
investments; also service rendered by reports of Dunn and 
Bradstreet. 

Arrange with a local banker to talk to the pupils about 
investments. Arrange with him to permit a group of chil- 
dren to have access to Moody’s ratings. Let them report 
the rating of some well-known and highly esteemed railroad 
or other good bond. Send another group to the bank to 
learn the rating on a bond of uncertain value. A written 
report should be required. 

5 Secure from newspapers or elsewhere accounts of dishonest 
investment schemes 


Situations for first half of eighth grade (optional) 


A Building a house 
I Figure all expenses: both interior and yard accessories, walks, 
streets etc: 
II Cost to include insurances of various kinds, as fire, walk, 
windows 
III Rents obtained 
1 If two flats unfurnished, with heating systems but no heat 
furnished 
2 If four small apartments furnished and heated etc. 
3 Whick is the better investment 


B Investing a gift of $500 in any way to secure greater income for — 
next 10 years 
I Income when used will furnish what part of living 
expenses ? | | 
II What would be value of same if income is invested each 
year? 
III How to invest that it shall be profitable 
IV Study of bonds and stocks; how to know whether reliable 
V When to buy? 


ELEMENTARY ARITHMETIC SYLLABUS 113 


VI When to sell? 
VII Why issued? 
VIII Why they fluctuate? 
IX Foreign bonds 
X Study of banks 
XI How banks invest? 
XII Mortgages 
XIII Real estate transfers 
XIV Savings and loan associations 


C Making an interest table 
D Making a tax table, using figures given by tax collector 


E Organization of a stock company — paying dividends, selling 
stock, etc. 


F Working problems in stocks and bonds, taking figures from stock 
reports in newspapers 


G Reading of gas and electric meters and bringing problems to 
school 


H Sending money by post office money order; filling out real blanks 
I Paying imaginary debts to one another by check 


J Making out real household accounts from bills brought by the 
children from home 


Eighth Grade (Second Half) 
A Drawing to scale 
I Why used? 
II By whom? 
III Accuracy in drawing absolutely essential 
IV Instruments used 
1 Real measurements and plans drawn. Measure school- 
room. Draw plan to show exact size of room, location 
of windows, blackboards and furniture 
2 Study map scales (more difficult work) 
3 Study simple plans of articles constructed in the shop. 
Close cooperation with teachers of manual training and 
sewing 
B The line. The path of a point in motion 
I Kinds 
1 Straight 
2 Curved 
3. Broken 


114 THE UNIVERSITY OF THE STATE OF NEW YORK 


The angle 

I Kinds 
1 Right 
2 Acute 
3 Obtuse 


C Construction 

I To bisect a straight line 

II To construct a perpendicular (1) at any point in a line, 

(2) at end of a line 

III To bisect an angle 

IV To construct equal angles 
V To construct a triangle equal to a given triangle 

VI To enlarge or reduce a rectangle or parallelogram 


D Graphs 

Gradually it has come about that people feel that a repre- 
sentation of facts by numbers conveys very little to the 
human mind. Pictures will often accomplish much more. 
Comparison is also a help. Hence has come the method of 
representing values by graphs. 

There are many ways of doing this. Any modern text 
will give some plan. Any study which gives the pupil 
ability to express values accurately and fully is good. 

There should also be a study to enable pupils to read 
graphs in order that they can gain from printed graphs the 
entire story they should tell. 


I Definition: the pictured form of presenting facts 


II Kinds 
iieine 
2 Bar 
3: Circular 
4 Pictographs 
III Essentials 
1 Accuracy 
2 Clearness 
3 Neatness 
4 Appealing to the eye and interest 
IV Things to note 
1 Any printing on the graph itself should be avoided, 
wherever possible 
2 All tabulations should be at the right side or below 


ELEMENTARY ARITHMETIC SYLLABUS 115 


3 Facts should be printed on another paper or far removed 
from the graph ; 
4 You should show clearly upon what basis you are repre- 
senting your facts 
V For illustrations of graphs see 
1 Line graph, plate 1, page 119 
2 Bar graph, plate 2, page 120 
3 Circular graph, plate 3, page 121 
Pictographs are really pictures used frequently by 
magazines. They should be of exact scale to tell the 
story truthfully. 
V1 There should be discussions to see what sort of graph will 
best illustrate the truth to be conveyed 
After a graph has been made the children should feel 
that it is really helpful in conveying the facts or the graph 
should be rejected. Practice on interpretation and tabula- 
tion from graphs. 
Coloring may be used to increase clearness. 

VII Much practice in constructing graphs. Abundant material 
for problems can be found in the World Almanac, census 
reports or statistical tables. Plot graphs of temperature 
and rainfall, school attendance. Two weeks or more may 
be profitably devoted to graphs. 

E The use of literal terms 

The work is meant to be in the nature of an easy 
approach to the subject. Only enough is given to awaken 
an interest in something quite new to the pupil. Too 
often the work in the eighth year proves an actual detri- 
ment to the pupil’s future work in algebra and many teach- 
ers of the algebra of the ninth year would much prefer 
that the pupils come to them entirely new to the subject. 
This work should be taken up slowly and each step should 
be thoroughly understood. Pupils who have done the work 
as outlined should be well prepared to undertake the diffi- 

culties of the fundamental operations. 

I Introduction to the use of symbols 

1 This work should be very simple in nature and should 
be designed to make the pupil familiar with the fact 
that quantities may be represented by letters or by 
letters combined with figures. Such quantities may be 
added, subtracted, multiplied and divided as with num- 
bers. The expression a@ means something mathemat- 


116 THE UNIVERSITY OF THE STATE OF NEW YORK 


ically exact when it appears in an algebraic combina- 
tion. Show that 3a is as easy a thing to consider as 
3 bushels, or 3 feet. The addition 3a-+ 2a = 5a is 
performed as easily as $3 + $2 = $5. Show also that 
a in the above expression may have any value that we 
may choose to give it 

2 The pupils should become familiar with the following: 
Coefficient, term (known and unknown), factor, ex- 
ponent, power, root 

3 Give pupils much practice in reading and writing very 
simple literal expressions as the work advances 

4 Simple literal terms. Let them not contain more than 
three letters. Deal only with small coefficients. It is 
recommended that there be no elaborate discussion of 
negative quantities. It is also recommended that the 
teacher present to the class only such combinations as 
will give positive results. 

F Addition of like terms. Practice in uniting terms so that the 
pupil shall not be confused with negative results. In other 
words, let the succession of terms be exactly the same as 
the pupil has had in numbers in arithmetic, for instance 
8a — 4a — 2a + 5a-+ a, ete. 

Note. If the teacher prefers, the negative quantity may 
be introduced and the regular work in algebraic addition may 
be taken up. The recommended procedure, however, has the 
advantage that addition in the ninth year will still be new to 
the pupil and the teacher will not therefore experience a lack 
of interest on the part of his pupils in that year. 

Show that 3a7b means 3 X a? & b and may be written 3.a?.b. 

G Simple numerical substitution. Secure simple material from any 
good text 

-H These axioms illustrated: addition, subtraction, multiplication, 
division. (Optional) The same roots of equals are equal. 
The same powers of equals are equal 

I The equation 
I Members of the equation 

II Transposition of terms and the axioms involved 
III Clearing of fractions with not more than two or three small 
denominators 
Note. The teacher may well approach the equation by 
the statement and solution of problems involving the use 
of X. Some teachers may prefer to give some simple 


ELEMENTARY ARITHMETIC SYLLABUS 117 


equations to perfect the mechanical manipulation before 
undertaking problem solution. 
J The solution of the problem 
I Read the problem carefully for the entire thought 
II Read the problem and determine what value is to be found 
first, or what value will make it possible to find all other 
values that may be required 
III Let X represent (equal) this value 
IV Read the problem again and with X as this value satisfy 
all other conditions in the problem 
V State the equation 
VI Solve the equation 
1 Clear of fractions if any are involved 
2 Transpose, if necessary, known terms to right side of 
equation and unknown to left 
3 Unite the terms. Equations involving negative members 
when simplified are to be avoided 
4 Divide both members of the equation by the coefficient 
of X 
5 Using the value of X find any other values required by 
the problem 
VII Check the work by substituting the value of X and see 
whether the answer is correct. It is also advisable to 
read the problem to see whether the answer satisfies all 
the conditions stated 


Notr. The teacher should make many problems for 
pupils to solve. Any easy text in algebra will furnish 
good material, and pupils may be encouraged to con- 
tribute original problems for class use. 


K The formula 
I The formation 
II The interpretation 
III The evaluation 


The teacher should in the beginning seek the material for the 
formula from the work in arithmetic previously studied. The 
formulas should be developed by the pupils themselves from what 
they already know. To illustrate, interest is found by multiplying 
what together? Principal, rate and time. 

Supposing that stands for principal, r for rate and ¢ for time, 
what will stand for interest? Interest (Int.) = prt. From what 


118 THE UNIVERSITY OF THE STATE OF NEW YORK 


the pupils already know about equations, they can develop from 
prt = Int. formulas as follows: 
Int. Int. Int. 


rt pt pr 

In commission if s = sales, y = rate of commission and c = com- 
mission, the class should develop c = rs. | 

Likewise develop formulas in profit and loss, insurance, taxes. 

Since the pupils know rules for areas of triangles, rectangles, 
parallelograms, trapezoids, let them derive their formulas, giving 
them letters to represent dimensions. 

Give practice on such formulas as D = rt. (Where D = distance, 
ry = rate of travel and ¢ the time traveled.) C = pn where C = cost, 
p = unit price and m = number purchased. 

Pupils should be given much practice in working out examples by 
use of formulas. 

Several of the more elementary books in algebra contain many 
simple formulas found in the lists of literal equations. Only the 
simplest of these should be used, but. much practice will not only 
interest the pupils but will strengthen them greatly for the work in 
algebra in the ninth year. 

Occasionally the teacher may well give a formula which is a state- 
ment of an arithmetical principle that the pupils know to test whether 
they can interpret the formula in terms of what they already know. 
Give considerable practice in the evaluation of simple formulas. 


ELEMENTARY ARITHMETIC SYLLABUS ~- 119 


PCATESL 


Deaths from Diphtheria in New York City 


1906 
1907 
1908 
1909 
1914 
19is 
1916 
{917 
1913 
i919 
1920 © 
iD2ZI 
[222 


, 1903 
, 1904 
1905 
1910 
Dit 
1912, 
1913 


2200 
2100 
2000 
i900 
IG0O 
I17OO 
[600 
1500 
1400 
1300 
1200 
1100 
1000 
960 
300 


From the above make a table showing yearly the number of deaths 
from diphtheria from 1903 to 1922 inclusive. 


What has been the general direction taken by the line? . 
Can a reason be given for the decrease shown by the graph? 
Find out what you can about the present treatment for diphtheria. 


120 THE UNIVERSITY OF THE STATE OF NEW YORK 


PruATEZ 


Cost of Public School Education in 1919-20 


238328333888 

gS Cee i Ge) 

esgssssssaas 

SssgseeRgsessRaggeg 
NEW YORK 106,045,319 
ono Ee pues Sees sat om 67,426,541 
CALIFORMIA ee cee 48,980,298 
NEW JERSEY 40,909,827 
MASSACHUSETTS GED 40,908,940 
COLORADO ae 13, 200,165 
VIRGINIA Pa 12,975,089 
ALABAMA me 9,118,691 


Compare the length of the bar representing Alabama’s school 
expenditure with that representing New York’s. 

Compare the population of the two states and determine which 
population is spending more for schools per capita. 

Determine the same for the other states. 

Which is spending more per capita of population, New Jersey or 


Massachusetts ? 


ELEMENTARY ARITHMETIC SYLLABUS yal 


RLATECO 


Distribution of Population in New York State in 1920 


25,000 — 
500,000 


Si hesS 
ABOVE 
500,000 


47-3 % 


K> % 
GI % * 


5,000 


RURAL AND 
VILLAGES 
LESS THAN 
2,500 

a7: 2%o 


Encourage pupils to tell in their own words the meaning of the 
above graph. 


How might the same graph drawn fifty years ago appear? 


What variation perhaps will be shown in the next twenty-five 
years? Give some reason for this. 


